Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

Research Article

Received 18 February 2013 Published online 5 August 2013 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.2942MOS subject classification: 35B41, 35B65

Infinite-energy solutions for the Cahn–Hilliardequation in cylindrical domains

A. Edena, V. K. Kalantarovb and S. V. Zelikc*†

Communicated by P. Colli

We give a detailed study of the infinite-energy solutions of the Cahn–Hilliard equation in the 3D cylindrical domains inuniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular poten-tials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, weprove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed toprove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and studyits properties. Copyright © 2013 John Wiley & Sons, Ltd.

Keywords: Cahn–Hilliard equation; unbounded domains; infinite-energy solutions

1. Introduction

We study the classical Cahn–Hilliard (CH) equations

@tuD�x.��xuC f .u/C g/ (1.1)

considered in an unbounded cylindrical domain � D R � ! (! is a smooth bounded domain) of R3 endowed with the Dirichletboundary conditions.

It is well known that the CH equation is central for the material sciences and extensive amount of papers are devoted to the mathe-matical analysis of this equation and various of its generalizations. From the physical point of view, the Neumann boundary conditionsare the most natural boundary conditions for (1.1), although the Dirichlet boundary conditions are also widely considered in the litera-ture especially in relation with the so-called viscous CH equations, see for example[1–4]. In particular, in the case where� is bounded,the analytic and dynamic properties of the CH equations are relatively well understood including the questions of well-posedness (evenin the case of singular potentials f ) and dissipativity, smoothness, existence of global and exponential attractors, and upper and lowerbounds for the dimension. We mention here only some contributors, namely, [4–20] (see also the references therein).

The situation in the case where the underlying domain is unbounded is essentially less clear even in the case of finite-energy solu-tions. Indeed, the key feature of the CH equation in bounded domains, which allows to build up a reasonable theory (especially in thecase of rapidly growing or singular nonlinearities) is the possibility to obtain good estimates in the negative Sobolev space W�1,2.�/,and to this end, one should use the inverse Laplacian .��x/

�1. But unfortunately, this operator is not good in unbounded domains(in particular, does not map L2.R3/ to L2.R3/), and this makes the most part of analytic tools earlier developed for the CH equationunapplicable to the case of unbounded domains. Thus, despite the general theory of dissipative PDEs in unbounded domains, whichseems highly developed nowadays (see the surveys [21] and [22] and references therein), the long-time behavior of solutions of theCH equation remains not well understood. Indeed, to the best of our knowledge, only the local results in this direction are availablein the literature, like the nonlinear (diffusive) stability of relatively simple equilibria (e.g.,kink-type solutions), relaxation rates to thatequilibria, asymptotic expansions in a small neighborhood of them and so on, see [23,24] and references therein. The situation becomeseven worse for more general infinite-energy solutions (e.g., for the initial data belonging to L1.Rn/ only). In this case, even the globalexistence of a solution is not known for the simplest cubic nonlinearity f .u/ D u3 � u (again, to the best of our knowledge) and theboundedness of solutions as t!1 is established only if f .u/ is linear outside of a compact in R, see [25].

aDepartment of Mathematics, Boðaziçi University, Bebek, Istanbul, TurkeybDepartment of Mathematics, Koç University, Rumelifeneri Yolu, Sariyer, 34450, Sariyer, Istanbul, TurkeycDepartment of Mathematics, University of Surrey Guildford, GU2 7XH UK*Correspondence to: S. V. Zelik, Department of Mathematics, University of Surrey Guildford, GU2 7XH UK.†E-mail: [email protected]

18

84

Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014, 37 1884–1908

A. EDEN, V. K. KALANTAROV AND S. V. ZELIK

This problem partially disappears if we consider the case where � is a cylindrical domain endowed by the Dirichlet boundary con-ditions (which is the main topic of the present paper). In this case, the inverse Laplacian is well defined (similar to the case of boundeddomains), and the theory of finite-energy solutions can be built straightforwardly combining the usual CH technique and the techniqueof weighted estimates [21, 26–31].

However, the class of finite-energy solutions is not satisfactory in unbounded domains (e.g., it does not contain physically importantsolutions, like spatially-periodic patterns and requires the additional strong restrictions on the potential f and external forces g) andshould be naturally replaced by the solutions in the so-called uniformly local Sobolev spaces, which typically have infinite-energy, seefor example the survey [21] for further discussion.

Thus, following the general strategy, it seems natural to consider the CH equation in the uniformly local phase spaces, and in order toobtain the reasonable estimates, we need to use the weighted energy estimates. But the application of this technique to the CH equa-tion is far from being straightforward even in the case of cylindrical domains because the presence of the weight prevents obtainingof the W�1,2-estimates. For this reason, the well-posedness of the CH equation in uniformly local spaces was known before only for theregular potentials with strong growth restrictions and only under the presence of the regularizing terms (the so-called microforces, see[32]) where the W�1,2-estimates are not necessary.

The aim of the present paper is to give a systematic study of the CH equations in cylindrical domains in the uniformly local phasespaces. To this end, we adapt some technique initially invented for the Navier–Stokes equations in cylinders, see [33] and [34], whichallow us to restore the crucial W�1,2-estimates for the reasonable classes of regular and singular potentials and to verify the well-posedness, dissipativity and existence of global attractors for these potentials. In particular, we are able to treat the regular potentialsof arbitrary polynomial growth (of course, under the standard dissipativity assumptions) as well as some classes of singular potentials.For instance, we prove the existence and uniqueness for the nonlinearities like

f .u/�u�

1� u2�� Ku (1.2)

with � � 5=3. Unfortunately, we are unable to verify the uniqueness for the most physical logarithmic potentials

f .u/� log1C u

1� u� Ku (1.3)

and for this reason, we will only construct later the trajectory attractor for the associated CH equation.The paper is organized as follows.In Section 2, we briefly recall the key facts related with the theory of weighted and uniformly local spaces, state and discuss our main

assumptions on the nonlinearity f and external forces g and give a precise definition for the solution of the CH equation.The dissipative estimates for the CH problem in the appropriate weighted and uniformly local Sobolev space will be derived in

Section 3.The central Section 4 is devoted to the uniqueness problem for the CH equations in uniformly local spaces. We also establish here

the smoothing property and the separation of the solutions from the singular points of the potential f .Then, in Sections 5 and 6, we study the attractors. We start (in Section 5) with the cases where the uniqueness is verified and prove

the existence of a ‘usual’ uniformly local attractor. In addition, we have also discussed here the additional features of that attractors,like, finite or infinite-dimensionality, and special solutions. After that (in Section 5), we turn to the case without uniqueness and verifythe existence of the so-called trajectory attractor in the weak topology of the trajectory phase space.

In Section 7, we adopt the general method presented in [35] to the CH problem and verify that any weak solution satisfies theweighted energy equality. Finally, based on that equality, we extend (in Section 8) the so-called energy method (see [36, 37], see also[38, 39]) to the case of uniformly local phase spaces and trajectory attractors and deduce the compactness of the attractor in a strongtopology as well as the attraction to it in the strong topology.

To conclude, we note that although we consider here only cylindrical domains � and only the case n D 3, most part of our resultscan be straightforwardly extended to any unbounded domain, which possesses the Friedrichs inequality, in particular, for a domainin space between two parallel planes

��DR2 � .0, 1/

�, and to the case of two spatial variables. However, the choice of the Dirichlet

boundary conditions is crucial for our method, and we do not know how to extend it to Neumann or periodic boundary conditions.

2. Assumptions and preliminaries

In the first part of this section, for the convenience of the reader, we recall the definitions and key ideas and properties of weightedand uniformly local spaces, which are crucial for our study of the CH equations (see [29, 33, 34, 40] for more detailed exposition). Inthe second part, we formulate the main assumptions on the nonlinear interaction function f , external force g, give the definition of asolution and so on.

We start with the class of admissible weights and the corresponding weighted spaces adapted to the case of cylindrical domains� :D R � !, x D .x1, x2, x3/ 2 � and ! is a bounded smooth domain of R2. Let ' 2 C.R/ be a weight function of exponential growthrate ˛ > 0. The latter means that

'.xC y/� Ce˛jxj'.y/, x, y 2R, '.x/ > 0 (2.1)


18

85


for some constant C > 0. Then, for every p 2 Œ1,1/, we define the space Lp'.�/with the following norm:

kukp

Lp'

:D

Z�'.x1/ju.x/j

p dx <1, (2.2)

where x1 2 R is the direction along the axis of the cylinder �. Analogously, for every l 2 N , we define the Sobolev space Wl,p' .�/ as a

space of distributions with derivatives up to order l inclusively belonging to Lp'.�/. The following standard Proposition (see, e.g., [29],

Theorems 1.1 and 1.2) is useful for what follows.

Proposition 2.1Let p 2 Œ1,1/ and let ' be a weight with exponential growth rate ˛ > 0. Then, for every ˇ > ˛, the following norms:

kukp1 :D

ZR'.s/kukp

Lp.�Œs,sC1�/ds and kukp

2 :D

ZR'.s/

ZR

e�ˇ jt�sjkukpLp.�Œt,tC1�/

dtds,

where�ŒT ,S� :D ŒT , S�� !, are equivalent to the usual weighted norm (2.2) in Lp' . Moreover, the equivalence constants depend only on

C and ˛ in (2.1) and ˇ > ˛ but are independent of the concrete choice of the weight '.

In particular, the second representation of the Lp' -norm allows us to obtain the estimates in general weighted spaces from the

analogous estimates for the specific exponential weights

�.x1/D �",s.x1/ :D e�"pjx1�sj2C1, (2.3)

where " > 0 and s 2 R are parameters (it is not difficult to check that the weights (2.3) satisfy (2.1) with ˛ D " and C D 1), and the first

representation of Proposition 2.1 gives the natural way to extend the definition of Wl,p' .�/ to the case of l 2R. Namely, for every l 2R,

we define the Sobolev space Wl,p' .�/ as a subspace of distributions for which the following norm is finite:

kukp

Wl,p'

:D

ZR'.s/kukp

Wl,p.�Œs,sC1�/ds. (2.4)

It is well known, see for example, [29, 33, 34], that this definition is self-consistent and, in particular (because of Proposition 2.1), forl 2N , it is equivalent to the definition given previously via the norm (2.2). Moreover, for l > 0, and 1< p <1, we have

W�l,p' .�/D

hWl,q

0,'.�/i�

,1

pC

1

qD 1,

whereh

Wl,q0,'.�/

i�is the dual of the space Wl,q

0,'.�/ (with respect to the weighted inner product of L2'.�/) and Wl,q

0,'.�/ is the closure

of C10 in the metric of Wl,q' .�/. In addition, exactly as in the non-weighted case, the fractional spaces Wl,p

' for the non-integer l can beobtained via the interpolation.

The next simple Proposition (see, e.g., [33] Proposition 2.14) is crucial for extending the embedding, interpolation and maximalregularity estimates to weighted spaces.

Proposition 2.2Let � be the weight function defined by (2.3). Then the multiplication operator

T� : u! �1=pu (2.5)

gives an isomorphism between the spaces Wl,p� .�/ and Wl,p.�/ (the non-weighted space!) for any l 2 R and 1 � p <1. Moreover, its

norm is independent of s 2R.

Propositions 2.1 and 2.2 allow to extend in an almost immediate way the classical elliptic and parabolic estimates to the class ofweighted spaces. Indeed, let us consider the simplest example of the Poisson equation in�with Dirichlet boundary conditions.

�xuD h, u j@� D 0 (2.6)

It is well known that this equation is uniquely solvable for any h 2 W�1,2.�/ and the Laplacian is an isomorphism between W1,20 .�/

and W�1,2.�/. Let h 2W�1,2� .�/. Then, introducing the new variable v D �1=2u and rewriting (2.6) in terms of v and using that

j�.n/.x/j � Cn"�.x/, n 2N , x 2R, "� 1, (2.7)

we end up with

�xv D �1=2hC "C".x1/vC "D".x1/@x1 v, v j@� D 0 (2.8)

18

86



for the appropriate smooth functions C" and D" such that

kC"kC1.�/CkD"kC1.�/ � C

and C is independent of "! 0. Thus, for small ", equation (2.8) is a perturbation of order " of the initial problem (2.6) and, therefore,is also uniquely solvable in the non-weighted spaces if " > 0 is small enough. Thus, combining this result with Proposition 2.2, we seethat the inverse Laplacian .��x/

�1 is well defined in the weighted spaces and

k.��x/�1hk2

W1,2�",s

.�/� Ckhk2

W�1,2�",s

.�/

if j"j � "0 > 0 and s 2 R is arbitrary. Moreover, the constant C in this estimate is independent of s 2 R. Therefore, multiplying thisestimate by '.s/ (arbitrary weight function of exponential growth rate ˛ < "0), integrating over s 2R and using the Proposition 2.1, weend up with the following result, see [33] for more detailed exposition.

Corollary 2.3There exists a positive constant ˛0 such that, for every weight function ' of exponential growth rate ˛ < ˛0, the inverse Laplacian.��x/

�1 with Dirichlet boundary conditions realizes an isomorphism between the spaces W�1,2' .�/ and W1,2

0,'.�/ and

C1khkW�1,2'� k.��x/

�1hkW1,20,'� C2khkW�1,2

', (2.9)

where the constants Ci are independent of the concrete choice of the weight ' satisfying (2.1).

Of course, the scheme described previously is restricted neither by the Hilbert case pD 2 nor by the Poisson equation and works for

any (uniformly) elliptic operator A in a Banach space Wl,p' .�/, 1 < p <1, see, for example, [33], Proposition 3.1. However, the assump-

tion that the operator A is invertible in the usual (non-weighted) Sobolev spaces (0 … �.A/ or at least 0 … �ess.A/) is crucial, so thisscheme is not applicable (at least in a straightforward way) to the Laplacian with Neumann boundary conditions where the essentialspectrum includes zero.

We now discuss the so-called uniformly local Sobolev spaces Wl,pb .�/, l 2 R, 1 � p <1, which are subspaces of distributions with

the following norm:

kukW

l,pb

:D sups2RkukWl,p.�Œs,sC1�/

<1. (2.10)

The properties of these spaces are studied in many papers, see [21, 34, 40] and references therein, so we state in what follows the keyproposition, which bridges the weighted and uniformly local spaces, see for example, [29], Corollary 1.6.

Proposition 2.4Let ' be the weight of the exponential growth rate ˛ such that ' 2 L1.R/ and let 's.x1/ :D '.x1 � s/, s 2 R, be the associated shiftedweight. Then the following norms are equivalent:

kukW

l,pb .�/

� sups2Rkuk

Wl,p's .�/

(2.11)

for all l 2 R and 1 � p < 1, and the equivalence constants depend on l, p,˛ and C from (2.1), but are independent of the concretechoice of the weight '.

Thus, if some regularity estimate is obtained in the weighted spaces of exponential growth rate with the constants, which are uni-form with respect to shifts of the weight, just taking the supremum with respect to shifts we end up with the analogous estimate inthe uniformly local spaces. This, together with the trick with reducing the weighted spaces to the non-weighted ones, explains whythe regularity theory in uniformly local spaces is very similar to the classical non-weighted one. We refer the reader to [29, 34, 40] formore details and only state here the result on the Poisson equation, which follows in a straightforward way from Corollary 2.3 and theequivalence given in (2.11).

Corollary 2.5The inverse Laplacian with Dirichlet boundary conditions realizes an isomorphism between W�1,2

b .�/ and W1,20,b.�/ (as usual, ‘0’ means

zero boundary conditions) and the analog of (2.9) for the uniformly local spaces holds.

We are now ready to return to the CH equation in the cylindrical domain��R3 (with the Dirichlet boundary conditions), which werewrite in the following form:

�@tuD�x�, � :D��xuC f .u/C g,u jtD0 D u0 , u j@� D �j@� D 0,

(2.12)

where uD u.t, x/ and�D �.t, x/ are the unknown order parameter and the chemical potential, respectively, g is a given external forceand f is a given nonlinearity.


18

87


We will consider two types of nonlinearities f : the regular and singular ones. Namely, for the regular case, we assume that8<:

1. f 2 C1.R,R/,2. f .u/u� �C, 8u 2R,3. f 0.u/��K , 8u 2R,

(2.13)

for some positive numbers C and K .For the singular case, we suppose that f is defined on the interval u 2 .�1, 1/ and8<

:1. f 2 C1.�1, 1/,2. limu!˙1f .u/D˙1,3. limu!˙1f 0.u/DC1

(2.14)

(which coincides with the conditions of [17] for the case of bounded domains). Of course, in that case, we should additionally assumethat

� 1< u.t, x/ < 1, foralmostall .t, x/ (2.15)

in order to make sense of the term f .u/. In the sequel, we will also need the function

f0.u/D f .u/C Ku (2.16)

which has the same behavior as u!˙1 in the regular case (and as u!˙1 in the singular case), but is monotone-increasing.Furthermore, we assume that the external force g 2 L2

b.�/ and the initial data u0 2ˆb with

ˆb :Dn

u0 2W1,2b .�/, F.u0/ 2 L1

b.�/o

, F.u/ :D

Z u

0f .v/dv. (2.17)

With a slight abuse of notation, we introduce the ‘norm’ in the (nonlinear) spaceˆb in the following natural way:

ku0k2ˆb

:D krxu0k2L2

b.�/CkF.u0/kL1

b.�/. (2.18)

We are now able to define a weak solution of the CH problem (2.12).

Definition 2.6A function u, is a (weak, infinite-energy) solution of problem (2.12) if�

1. u 2 L1.RC,ˆb/\ C�Œ0,1/, L2

loc.�/�

,2. f .u/,�xu,rx� 2 L2

b.RC ��/(2.19)

and the equations (2.12) are satisfied in the sense of distributions. In the case of singular f , we additionally assume that (2.15) is sat-isfied. Here and in what follows, L2

loc.�/ stands for the Frechet space generated by seminorms kukL2.�Œs,sC1�/, s 2 R (and the Frechet

spaces Wl,ploc.�/ are defined analogously), L1.RC,ˆb/ simply means thatrx u and F.u/belong to L1

�RC, L2

b.�/�

and L1�RC, L1

b.�/�,

respectively, and, in a complete agreement with (2.10), the norm in L2b.RC ��/ is defined by

kukL2b.RC��/

:D supT2RC

sups2RkukL2

b.ŒT ,TC1��Œs,sC1�/.

Remark 2.7The typical example of the regular nonlinearity f is

f .u/D u3 � Ku, K > 0 (2.20)

which corresponds to the classical CH equation or, more generally, any polynomial of the odd degree and positive leading term. Forthe singular case, the most physical is the so-called logarithmic nonlinearity (1.3), although the nonlinearities with stronger algebraicsingularities are also widely used, see [17] and references therein.

Note that in all these cases, f satisfies the second dissipativity condition (2.13) with some positive constant C and K (the case K � 0in (2.20) or in (1.3) is not interesting for applications to phase transitions because it leads to the monotone equation, which can exhibitonly one phase). In that case, even with g � 0, the CH equation does not possess a global attractor in the usual Sobolev spaces (say,L2.�/), see [21] for further discussion.

Nevertheless, there is a bit artificial (but interesting) class of non-monotone nonlinearities f , which satisfy

f .u/u� 0, u 2R. (2.21)

In that case (analogous to [28, 30], see also [21]), there is a global finite-dimensional attractor for the associated CH problem in L2.�/

if the external forces belong to L2.�/ and the situation is very similar to the case of bounded domains. In the present paper, we aremainly interested in the phenomena essentially related to unbounded domains, so we will not pay any extra attention to that particularcase (see, however, Corollary 5.5).

18

88



3. A priori estimates and the existence of solutions

In this section, we prove the existence of solutions for the CH problem as well as their further regularity and key dissipative estimates.Actually, for most of the results of this section, the difference between the singular and regular cases is not essential, so we will treatthem from a unified point of view. Note also that to simplify the notation, different constants may be denoted by the same symbol C(or Ci , ˛, etc.) if this does not lead to a confusion.

The main result of this section is the following theorem.

Theorem 3.1Let the nonlinearity f satisfy assumption (2.13) or (2.14), and the external force g 2 L2

b.�/. Then, for any u0 2 ˆb, there exists at leastone solution u.t/ of the problem (2.12) (in the sense of Definition 2.6), which satisfies the following dissipative estimate:

ku.T/k2W1,2

b .�/CkF.u.T//kL1

b.�/Ck�xuk2

L2b.ŒT ,TC1��/

C

Ckf .u/k2L2

b.ŒT ,TC1��/Ckrx�k

2L2

b.ŒT ,TC1��/� Cku0k

2ˆb

e�˛T C C

�kgk2

L2b.�/C 1

� (3.1)

where the positive constants C and ˛ are independent of T and u0.

ProofWe first give the formal derivation of the dissipative estimate (3.1) (some explanations on how to justify it and to verify the exis-tence of a solution will be given afterwards). To this end, we multiply in L2.�/ equation (2.12) by �.x1/�, where the weight function�.x1/ D �",s.x1/ is defined by (2.3), s 2 R and "0 is a sufficiently small parameter which will be specified later. Then, after thestraightforward transformations, we arrive at

d

dt.1=2krxu.t/k2

L2�

C .F.u.t//,�/C .u.t/�, g//Ckrx�.t/k2L2�

C [email protected]/,�0@x1 u.t//C .@x1�.t/,�0�.t//D 0

(3.2)

(here and in what follows, we denote by ., / the usual inner product in the Hilbert space L2.�/). Using the Poincaré inequality togetherwith inequality (2.7), the last term in the left-hand side of (3.2) can be estimated as follows:

2j.@x1�,�0�/j D j��00,�2

�j � C"k�k2

L2�

� C0"krx�k2L2�

. (3.3)

Furthermore, from the equation (2.12), we conclude that

j.@tu,�0@x1 u/j D .�x�,�0@x1 u/� C"

�krx�k

2L2�

Ckuk2W2,2�

�. (3.4)

In order to close the estimate, we only need to use the weighted maximal regularity estimate for the elliptic equation

�xu� f .u/D h, u j@� D 0. (3.5)

Lemma 3.2Let u 2 W1,2

b .�/ be a solution of equation (3.5) with h 2 L2b.�/ and the nonlinearity f satisfies assumptions (2.13) or (2.14). Then,

u 2W2,2b .�/, and the following estimate holds:

kuk2W2,2�

Ckf .u/k2L2�

C .�jf .u/j, juj/� Ckhk2L2�

C C", (3.6)

where � D �",s.x1/, the constant C is independent of " > 0 being small enough and the constant C" depends on " (irrespective of thetype of the nonlinearity f , singular or regular).

ProofAlthough the regularity estimate (3.6) for the semilinear equation (3.5) is well known (see, e.g., [29]), for the convenience of the reader,we give its derivation later. To this end, we first multiply equation (3.5) by �u. Then, using that in both cases, the inequality f .u/u � �Cholds; after the standard transformations involving (2.7) and the Poincare inequality, we arrive at

krxuk2L2�

C .�jf .u/j, juj/� Ckhk2L2�

C C". (3.7)

After that, we may multiply equation (3.5) by rx.�rxu/ and use that f 0.u/ � �K , which gives again after the standard transformationsthat

k�xuk2L2�

� .CC K/kuk2W1,2�

Ckhk2L2�

. (3.8)


18

89


This estimate together with (3.7) and the weighted L2 ! W2,2-regularity estimate for the Laplacian gives the desired estimate (3.6).This ends the proof of the lemma. �

Applying Lemma 3.2 to the equation

�xu� f .u/D g�� (3.9)

using now estimates (3.3), (3.4) and (3.6) and fixing " small enough, we deduce from (3.2) that

d

dt

�1

2krxu.t/k2

L2�

C .F.u.t//,�/C .�u.t/, g/

�Ckrx�.t/k

2L2�

� C

�1Ckgk2

L2�

�, (3.10)

where " is now fixed and the constant C is independent of the parameter s in the definition of the weight �.Finally, using Lemma 3.2 together with the fact that F.u/� jf .u/jjuj C Cjuj2, we conclude that

krxu.t/k2L2�

C .F.u.t//,�/� C

�krx�.t/k

2L2�

Ckgk2L2�

C 1

�

and, therefore, inequality (3.10) can be rewritten in the form

d

dt

�1=2krx u.t/k2

L2�

C .F.u.t//,�/C .u.t/�, g/

�C

C ˛

�1=2ku.t/k2

L2�

C .F.u.t//,�/C .u.t/�, g/

�C

C

�krx�.t/k

2L2�

Cku.t/k2W2,2�

Ckf .u.t//k2L2�

�� C

�1Ckgk2

L2�

�,

(3.11)

for some positive constants ˛, and C, which are independent of u and t. Applying now the Gronwall inequality to (3.11), we obtainthe weighted analog of the desired dissipative estimate (3.1):

ku.T/k2W1,2� .�/

CkF.u.T//kL1�.�/

Ck�xuk2L2�.ŒT ,TC1��/

Ckf .u/kL2�

�ŒT , T C 1�� ukL2

��

�C

Ckrx�k2L2�.ŒT ,TC1��/

� C

�ku0k

2W1,2�

CkF.u0/kL1�

�e�˛T C C

�kgk2

L2�.�/

C 1

� (3.12)

for some positive constants ˛ and C. In addition, these constants are independent of the parameter s in the weight function � D �",s.Therefore, taking the supremum on s 2 R from the both sides of inequality (3.12) and using (2.11), we obtain the desired dissipativeestimate (3.1) in the uniformly local spaces.

Thus, it only remains to verify the existence of a solution. In fact, it can be carried out in many standard ways. In particular, one of thesimplest ways is to approximate the initial data u0 and the external force g by a sequence un

0 and gn of smooth functions with compactsupport. Then, for the approximate CH problems (2.12) with those data, the usual unweighted theory is applicable and the existenceand uniqueness of a solution un.t/ can be verified exactly as in the case of bounded domains, see [6] or [17] for the details. Thus, argu-ing as before, we obtain the dissipative estimate (3.1) for the approximate solutions un.t/ with constants ˛ and C independent of n.Passing after that to the limit n!1, we end up with the desired solution and the desired dissipative estimate (3.1) in the uniformlylocal spaces. �

We now formulate one more regularity result, which will be useful for verifying the uniqueness of solution in the case of singularpotentials.

Corollary 3.3Let the assumptions of the Theorem 3.1 hold, g 2 L6

b.�/, and let u be a solution of the problem (2.12) constructed in that theorem.Then, the following estimate holds:

sups2Rkf .u/kL2.Œt,tC1�,L6.�Œs,sC1�//

� Cku0kˆb e�˛t C C�

1CkgkL2b

�, (3.13)

where the positive constants C and ˛ are independent of t and u.

ProofIndeed, without loss of generality, we may assume that f 0.u/� 0. Then, multiplying the equation (3.9) by �f .u/5 and using (2.7) and theHölder inequality with exponents 6 and 6=5, we arrive at

5��f 0.u/f .u/4rxu,rxu

�Ckf .u/k6

L6�

D�� g,�f .u/5

��

��0f .u/5, @x1 u

�� Ckf .u/k5

L6�

�k�kL6

�CkgkL6

�Ckrx ukL6

�

��

� 1=2kf .u/k6L6�

C C0�krxuk6

L6�

Ckgk6L6�

Ck�k6L6�

�.

(3.14)

18

90



Therefore,

kf .u/k2L6�

� 2C0�kgk2

L6�

Ck�k2L6�

Ckrxuk2L6�

�.

Integrating this inequality with respect to Œt, t C 1�, taking the supremum over s 2 R and using the embedding W1,2 � L6 and theLemma 3.2, we finally have

sups2Rkf .u/kL2.Œt,tC1�,L6.�Œs,sC1�//

� C�kgkL6

bCkrx�kL2

b.Œt,tC1��/

�.

This estimate together with the dissipative estimate (3.1) gives (3.13) and finishes the proof of the corollary. �

We recall that up to now, we consider only the solutions of equation (2.12) with sufficiently regular initial data u0 2ˆb. However, it iswell known that at least in the case of bounded domains, the initial boundary value problem for the CH equation is well-posed for lessregular initial data u0 2W�1,2.�/ and that the W�1,2-estimates of solutions are crucial for the theory of this equation, see [4, 17, 41]. Inthe case of unbounded domains and uniformly local phase spaces, the situation still becomes more complicated and more delicate, aswe will see later, but it is still possible to verify the existence of solutions for u0 2 W�1,2

b .�/ in the general case of regular and singularpotentials.

Theorem 3.4Let the nonlinearity f satisfy conditions (2.13) (regular case) together with the polynomial growth restriction

f .u/u��CC C1jujpC1, jf .u/j � C2

�1C jujp

�(3.15)

for some positive constants C, C1, C2 and some exponent p > 0, or (2.14) (singular case) and u0 2 W�1,2b .�/ (in the case of singular

potentials, we should assume, in addition, that ju0.x/j < 1 almost everywhere). Then, there exists at least one solution u.t/ of theproblem (2.12), which becomes more regular (u.t/ 2ˆb) for all t > 0 and satisfies all assumptions of Definition 2.6 for t > 0. Moreover,

ku.t/k2W�1,2

b .�/Ckrx uk2

L2b.Œt,tC1��/

C

Ckf .u/ukL1b.Œt,tC1��/ � Cku0k

2W�1,2

b .�/e�˛t C C

�1Ckgk2

L2b

�,

(3.16)

where the positive constants C and ˛ are independent of u and t. In addition, the following smoothing estimate holds:

ku.t/kˆb � Ct�1=2ku0kW�1,2bC C

�1CkgkL2

b

�, t � 1 (3.17)

for some positive constant C and the estimate (3.1) holds for t > 0 with u0 replaced by u.t/.

ProofAs it is carried out before, we will only verify estimates (3.16) and (3.17) in what follows. The existence of a solution can be obtained, forexample, by approximating the non-regular initial data u0 by the regular ones un

0 2 ˆb such that kunkW�1,2b .�/

are uniformly bounded

and un0 ! u0 in W�1,2

loc .�/. Then, the existence of a solution un.t/ with un.0/D un0 follows from Theorem 3.1, and (3.16) and (3.17) give

the uniform bounds for the sequence un.t/. In particular, because of the immediate smoothing property (3.17), we have the uniformbound (3.1) for the sequence un with respect to n for every fixed T > 0. Thus, the passage to the limit n!1 can be carried out exactlyas in Theorem 3.1 (see again [6] of [17]). So from now on, we concentrate on the formal derivation of the key a priori estimates (3.16)and (3.17).

Let v.t/ :D .��x/�1u.t/, and let us rewrite the problem (2.12) in the equivalent form:�

@tv D�xu� f .u/� g, x 2�, t > 0,u.0/D u0, u j@� D 0.

(3.18)

Multiplying this equation by�rx.�".x/rxv/ and integrating over x 2�, we obtain

1

2

d

dtkrxvk2

L2�

Ckrx uk2L2�

C .�f .u/, u/DD�.@x1 u,�0@x1 u/

C .rxu,rx.�0@x1 v//C .f .u/,�0@x1 v/C .g,rx.�rxv//.

(3.19)

We now use the inequality (2.7) together with the maximal regularity of the Laplacian in weighted spaces (analogously to Corollary 2.3)

C2krx ukL2�� kvkW3,2

�� C1krxukL2

�(3.20)

(recall that � D �",s is given by (2.3), " is small enough and s 2R is arbitrary). Then, the formula (3.19) reads

1

2

d

dtkrxv.t/k2

L2�

C ˛kvk2W3,2�

C .�jf .u/j, juj/� C"

�1Ckgk2

L2�

�C C".�jf .u/j, jrxvj/, (3.21)


18

91


where the positive constants C and ˛ are independent of "! 0.Thus, we only need to estimate the last term in the right-hand side of (3.21). Let us consider the regular and singular cases separately.

Assume that f is regular and the assumptions (3.15) hold. Then, using the maximal LpC1-regularity for the Laplacian in the weightedspaces

krxvkL

pC1�

� CkukW�1,pC1�

� C1kukL

pC1�

together with Hölder inequality and assumption (3.15), we arrive at

j.f .u/,�jrxvj/j � Ckf .u/kLq�krx vk

LpC1�

� C

�"�1CkukpC1

LpC1�

�, (3.22)

where 1q C

1pC1 D 1 and the constant C is independent of ". Using again inequalities (3.15) and fixing " > 0 small enough, we finally

deduce that

d

dtkrxv.t/k2

L2�

C ˛kv.t/k2W3,2�

C ˛ku.t/k2W1,2�

C .�jf .u.t//j, ju.t/j/� C"

�1Ckgk2

L2�

�. (3.23)

Let us now consider the singular case. The situation here is even simpler as we a priori know that ku.t/kL1 � 1; therefore, according tothe regularity of the Laplacian in the uniformly local spaces, we conclude that

krxv.t/kL1.�/ � C. (3.24)

Therefore, the second term in the right-hand side of (3.21) can be estimated by

j.f .u/,�jrxvj/j � Ckf .u/kL1�

and, because jf .u/j � 2f .u/uC C1, we again may fix " small enough in such a way that the last term in the right-hand side of (3.21) willbe controlled by the last term in the left-hand side. Thus, in this case, we also arrive at the inequality (3.23).

After obtaining the differential inequality (3.23), it is not difficult to deduce the desired estimates (3.16) and (3.17) and finish the proofof the theorem. To this end, we note that the Gronwall inequality applied to (3.23) gives

kv.T/k2W1,2�

C

Z TC1

Tku.t/k2

W1,2�

C .�jf .u.t//j, ju.t/j/dt �� Ckv.0/k2W1,2�

e�˛T C C

�1Ckgk2

L2�

�. (3.25)

In order to deduce the estimate (3.16) from (3.25), it is enough to remind that � D �",s and take the supremum over s 2 R from theboth sides of that inequality (analogous to the derivation of (3.1) from (3.12)).

Let us verify now the smoothing property (3.17). To this end, we multiply inequality (3.11) by t and integrate with respect to t 2 Œ0, 1�.That gives,

tku.t/k2W1,2�

C t.F.u.t//,�/� C

Z 1

0ku.s/k2

W1,2�

CkF.u.s//kL1�

dsC C

�1Ckgk2

L2�

�. (3.26)

Using now the inequality (3.25) for estimating the right-hand side of (3.26) and using the inequality F.u/ � f .u/uC Cjuj2, we arrive atthe weighted analog of the estimate (3.17). Taking finally the supremum over s 2 R, we derive the uniformly local estimate (3.17) andfinish the proof of the theorem. �

Remark 3.5The proof of the last theorem indicates the main difference of the CH equation theory in weighted spaces in comparison with theclassical unweighed case, namely, the presence of the additional term C".jf .u/j,�jrxvj/ in the W�1,2-estimate (3.21). This term is notsign-defined and factually destroys the global Lipschitz continuity of the solution semigroup in weighted spaces, which is the maintechnical tool for handling the CH equation with singular or fast growing potentials. In the previous theorem, we were able to over-come this difficulty and obtain the dissipative W�1,2-estimate in the weighted and uniformly local spaces in almost the same form asfor the non-weighted case. However, as we will see in the next section, this difficulty will force us to put more restrictive assumptionson f if we want to establish the uniqueness.

4. Uniqueness and regularity

In this section, we pose some additional restrictions on the nonlinearity f , which will allow us to establish the uniqueness of solutionsand some crucial smoothing effects for the CH equation in the uniformly local spaces. We start with the case of regular potentials andassume also that the following is true:

jf 0.u/j � C1F.u/C C2, (4.1)

18

92



for all u 2 R and for some fixed positive C1 and C2. Note that (4.1) does not look as a big restriction for the case of regular potentials(although it excludes completely the case of singular f , which will be separately discussed later). In particular, the polynomial nonlin-earities (see condition (3.15)) always satisfy this assumption. The next theorem gives the uniqueness of a solution for the CH equationwith regular potentials.

Theorem 4.1Let the assumptions of Theorem 3.1 hold and let, in addition, f be regular and (4.1) holds. Then, for every u0 2 ˆb, the solution u.t/constructed in the Theorem 3.1 is unique and, for every two such solutions u1.t/ and u2.t/, the following weighted Lipschitz continuityin W�1,2

�",sholds:

ku1.t/� u2.t/kW�1,2�",s� CeKtku1.0/� u2.0/kW�1,2

�",s, (4.2)

where the weights �",s are defined by (2.3) and the constants C and K depend on theˆb-norms of the initial data u1.0/ and u2.0/, butindependent of " 1 and s 2R.

ProofLet w.t/ :D u1.t/� u2.t/ and let v.t/ :D .��x/

�1w.t/. Then, these functions solve

@tv D�xw � l.t/w, (4.3)

where l.t/ :D f.u1.t//�f.u2.t//u1.t/�u2.t/

DR 1

0 f 0.u1.t/C .1� /u2.t//d.

Multiplying the equation (4.3) by rx.�rxv/ (note that all terms have sense because u1 and u2 are solutions in the sense ofDefinition 2.6) and using the inequality l.t/� �K , we obtain

.Œl.t/C K�w,�0@x1 v/� 1=2.�Œl.t/C K�w, w/C C"2.�Œl.t/C K�@x1 v, @x1 v/,

we end up with the following estimate (similar to (3.19)):

d

dtkvk2

W1,2�

C 2˛kvk2W3,2�

� K

�kvk2

W1,2�

Ckvk2W2,2�

�C C"2.�l.t/rxv,rxv/, (4.4)

where C is independent of " being small enough. As usual, the term with the W2,2-norm of v can be easily estimated by interpolationbetween W3,2 and W1,2, so we only need to estimate the last term on the right-hand side of (4.4). Note also that up to this moment, wehave nowhere used the additional restriction that f is regular and satisfies (4.1).

In order to estimate the term l.t/, we remind that the function F.u/ C Ku2=2 is convex (as f 0.u/ � �K). Consequently, because ofcondition (4.1),

jf 0.u1C .1� /u2/j � C1F.u1C .1� /u2/C C2 � C1.F.u1/C F.u2//C C2C K�

u21C u2

2

�.

As F.w/ 2 L1b for all w 2ˆb, from (2.19), we conclude that

kl.t/kL1b.�/� C, t 2RC, (4.5)

where the constant C is independent of t. Thus, applying the Hölder inequality with exponents 1 and1 to the right-hand side of (4.4)and using (4.5), we obtain

d

dtkvk2

W1,2�

C ˛kvk2W3,2�

� Ckvk2W1,2�

C C"2krx vk2L1�1=2

. (4.6)

We estimate the last term in (4.6) by using the interpolation inequality

krx vk2L1�1=2� Ckvk1=2

W1,2�

kvk3=2

W3,2�

and obtain from (4.6) the following inequality

d

dtkvk2

W1,2�

C ˛=2kvk2W3,2�

� Lkvk2W1,2�

, (4.7)

where the constant L is independent of the shift parameter s in the definition of � D �",s. Applying the Gronwall inequality to (4.7) andusing the weighted W�1,2

� !W1,2� regularity of the Laplacian, we end up with (4.2) and finish the proof of the theorem. �

Corollary 4.2Under the assumptions of Theorem 4.1, the CH problem (2.12) generates a dissipative semigroup S.t/ in the phase spaceˆb:

S.t/ :ˆb!ˆb, S.t/u0 :D u.t/, (4.8)


18

93


where u.t/ solves (2.12) with u.0/D u0. Moreover, this semigroup is locally Lipschitz continuous in the W�1,2b -norm:

kS.t/u1 � S.t/u2kW�1,2b .�/

� CeLtku1 � u2kW�1,2b .�/

, (4.9)

where the constants C and L depend only on theˆb-norms of u1 and u2.

Indeed, in order to verify (4.9), it suffices to take the supremum over the shift parameter s 2 R from both sides of (4.2), and theremaining assertions are immediate corollaries of Theorem 4.1.

We now discuss the further regularity and smoothing property for the solutions of the CH equation with regular potentials in theuniformly local spaces.

Corollary 4.3Let the assumptions of Theorem 4.1 hold. Then, for every t > 0, u.t/ 2W2,2

b .�/, and the following estimate is valid:

ku.t/k2W2,2

b .�/[email protected]/k2

W�1,2b .�/

C sup�2R

Z tC1

[email protected]/k2

L2.�Œ�,�C1�/ds�

� CtC 1

tku0k

2ˆb

e�˛t C C

�1Ckgk2

L2b.�/

�,

(4.10)

where the positive constants C and ˛ are independent of t.

ProofIndeed, let us formally differentiate the equation (2.12) with respect to t (because the solution is unique, this action can be easily justi-fied by the appropriate approximation procedure) and denote wD @tu and v D .��x/

�1@tu. Then, these functions solve the analog ofthe equation (4.3):

@tv D�xw � f 0.u.t//w. (4.11)

Multiplying this equation by rx.�rxv/ and arguing exactly as in the proof of Theorem 4.1, we obtain the estimate (4.7). Multiplyingthat estimate by t and integrating over Œ0, t�, we have

tkrxv.t/k2L2�

� C

Z t

0krxv.t/k2

L2�

dt, t � 1. (4.12)

Because, according to equation (2.12), v.t/D �.t/, the right-hand side of (4.12) can be estimated using (3.12). That gives

[email protected]/k2W�1,2�

� C

�ku0k

2W1,2�

CkF.u0/kL1�Ckgk2

L2�.�/

C 1

�, t � 1. (4.13)

The estimate (4.13) together with the dissipative estimate (3.1) and the trick with taking supremum over the shift parameter give thedesired estimate (4.10) for @tu. In order to obtain the desired estimate (4.10) for the W2,2

b -norm, it suffices now to use the Lemma 3.2.Corollary 4.2 is proved. �

Thus, because of the embedding W2,2b .�/ � Cb.�/, the regular nonlinearity f becomes subordinated to the linear part of the equa-

tion (no matter how fast f grows), and obtaining the further regularity of solutions reduced the standard bootstrapping procedure byusing the highly developed weighted theory for the linear equations. We do not discuss that standard thing here, and the rest of thesection will be devoted to more interesting and more complicated case of singular nonlinearity f . In that case, we have to pose ratherrestrictive assumption that there exists a convex function R : .�1, 1/!R such that

˛2R.u/� C1 � jf .u/j � ˛1R.u/C C1, jf 0.u/j � ˛3jf .u/j8=5C C3 (4.14)

for some positive ˛i , i D 1, 2, 3. Roughly speaking, condition (4.14) means that the singularities of the function f at u D ˙1 aresufficiently strong. In particular, the function

f .u/Du�

1� u2�� Ku

satisfies that assumption if and only if � � 5=3. Unfortunately, we are unable to handle the most physical case of logarithmic potential

f .u/D log1C u

1� u� Ku (4.15)

and we do not know whether or not the uniqueness result holds for this nonlinearity.The following theorem gives the analog of the uniqueness result of Theorem 4.1 for the case of sufficiently strong singular potentials.1

89

4



Theorem 4.4Let the nonlinearity f be singular and satisfy assumptions (2.14) and (4.14). Let also g 2 L6

b.�/. Then, for every u0 2 ˆb, the solution (inthe sense of Definition 2.6) of equation (2.12) is unique and estimate (4.2) holds for any two solutions of that equation (with differentinitial data).

ProofLet, as in Theorem 4.1, u1.t/ and u2.t/ be two solutions of the CH equation, w.t/ :D u1.t/� u2.t/ and v.t/ :D .��x/

�1w.t/. Then, thesefunctions satisfy equation (4.3). In addition, because of assumption (4.14), we have

jl.t/j � C�jf .u1.t//j

8=5C jf .u2.t//j8=5C 1

�which, together with Corollary 3.3, gives

supt2RC , s2R klkL5=4.Œt,tC1�,L15=4.�Œs,sC1�//� C D Cu1,u2 . (4.16)

Moreover, arguing as in the proof of Theorem 4.1, we obtain the differential inequality (4.4). As we will see later, the regularity estimate(4.16) suffices to control the additional weighted term and to close the uniqueness estimate. However, because (4.16) does not imply

that l 2 L5=4�Œt, tC 1�, L15=4

b .�/�

, we should proceed in a more accurate way. Namely, integrating the inequality (4.4) over t, we end up

with

krxvk2L1.Œ0,t�,L2

�",s/C ˛

Z t

0kv./k2

W3,2�",s

d �

� L

Z t

0kv./k2

W1,2�",s

d Ckrxv.0/k2L2�",s

C C"2Z t

0

�jl./j,�",sjrxv./j2

�d

(4.17)

(because the dependence of �",s on " and s is crucial for the proof, for the sake of clarity, we will explicitly indicate it in what follows tillthe end of this proof ). It is important to note that the positive constants C, L and ˛ are independent of "! 0. In order to estimate thelast term in (4.17), we use the following inequalities

C�1ZR�",s./kvkL1.�Œ�,�C1�/

d � kvkL1�",s� C

ZR�",s./kvkL1.�Œ�,�C1�/

d, (4.18)

where the constant C is independent of "! 0 (Proposition 2.1). Assuming without loss of generality that t � 1 and using the inequality(4.18) together with the control (4.16) and Hölder inequality, we haveZ t

0

�jl./j,�",sjrxvj2

�d � C

ZR�",s./kl jrxvj2kL1.Œ0,t��Œ�,�C1�/

d �

� C

ZR�",s./klkL5=4.Œ0,t�,L15=4.�Œ�,�C1�//

krx vk2L10.Œ0,t�,L30=11.�Œ�,�C1�//

d �

� C1

ZR�",s./krxvk2

L10.Œ0,t�,L30=11.�Œ�,�C1�//d.

(4.19)

Using now the following interpolation inequality

kzkL10.Œ0,t�,L30=11/ � Ckzk4=5L1.Œ0,t�,L2/

kzk1=5L2.Œ0,t�,W2,2/

with the constant C independent of t � 1 (we recall that the space dimension nD 3), we may continue the estimate (4.19):

C"2Z t

0

�jl./j,�",sjrx vj2

�d �

� C2

ZR�",s./

�"5=2krx vk2

L1.Œ0,t�,L2.�Œ�,�C1�//

�4=5 �krxvk2

L2.Œ0,t�,W1,2.�Œ�,�C1�//

�1=5d �

� ˛

Z t

0kv./k2

W3,2�",s

d C C"5=2ZR�",s./krxvk2

L1.Œ0,t�,L2.�Œ�,�C1�//d.

(4.20)

However, the last term on the right-hand side of (4.20) still cannot be estimated by the first term on the left-hand side of (4.17), and wehave only the obvious one-sided estimate:

krx vk2

L1�Œ0,t�,L2

�",s

� � C

ZR�",s./krxvk2

L1.Œ0,t�,L2.�Œ�,�C1�//d. (4.21)

In order to overcome this difficulty, we recall that �".x/D �",s.x/� e�" if x 2�Œs,sC1�. Therefore,

krx vkL1.Œ0,t�,L2.�Œs,sC1�//� Ckrxvk2

L1�Œ0,t�,L2

�",s

�.


18

95


This estimate together with (4.20) and (4.17) gives the following estimate:

krxvk2L1.Œ0,t�,L2.�Œs,sC1�//

� C

�krxv.0/k2

L2�",s

C

Z t

0krxv./k2

W1,2�",s

d

�C

C C"5=2ZR�",s./krxvk2

L1.Œ0,t�,L2.�Œ�,�C1�//d,

(4.22)

where t � 1 and the constant C is independent of s 2R and "! 0.We are now ready to close the uniqueness estimate. To this end, we multiply estimate (4.22) by �"=2,l.s/, where l 2 R is a new shift

parameter and integrate over s 2R. Then, using the estimateZR�"=2,l.s/

ZR�",s./z./d ds� C"�1

ZR�"=2,l./z./d, (4.23)

see Proposition 2.1, we arrive at

ZR�"=2,s./krxvk2

L1

�Œ0, t�, L2.�Œ�,�C1�/

�, d �C"�1

krxv.0/k2

L2�"=2,s

C

Z t

0krxv./k2

W1,2�"=2,s

&, d

!C

C C"3=2ZR�"=2,s./krxvk2

L1.Œ0,t�,L2.�Œ�,�C1�//d.

(4.24)

Fixing here " > 0 to be small enough (say, C"3=2 D 1=2) and using (4.21), we conclude that

krxv.t/k2L2�"=2,s

� Ckrxv.0/k2L2�"=2,s

C C

Z t

0krx v./k2

L2�"=2,s

d .

This estimate together with the Gronwall inequality gives the desired estimate (4.2) and finishes the proof of the theorem. �

The following corollary is a straightforward analog of Corollaries 4.2 and 4.3 for the singular case.

Corollary 4.5Let the assumptions of Theorem 4.4 hold. Then, the problem (2.12) generates a dissipative semigroup (4.8) in the phase space ˆb,which is locally Lipschitz continuous in it (i.e., (4.9) holds). Moreover, any solution of (2.12) possesses the smoothing property (4.10).

Indeed, these assertions follow from Theorem 4.4 exactly as in the regular case, so we omit their proofs here.Let us mention that in contrast to the regular case, the proved W2,2

b -regularity (4.10) of solutions is not sufficient to apply the furtherregularity using the linear theory because we still have the singular term f .u/ and need to prove that the solution cannot reach thesingular points uD˙1. The next proposition gives such a result.

Proposition 4.6Let the assumptions of Theorem 4.4 hold. Then, f .u.t// 2 L1.�/ for all t > 0 and

kf .u.t//kL1.�/ �1C tN

tNe�˛tQ.ku0kˆb/CQ

�kgkL6

b.�/

�, (4.25)

where ˛ and N are some positive constants and Q is a monotone function, which are independent of t and u0.

ProofThe proof of this result is analogous to [21], and the fact that the underlying domain � is unbounded does not make any difference.Nevertheless, for the convenience of the reader, we give in what follows a schematic derivation of the desired estimate. First, accordingto (4.10) and the embedding W1,2 � L6,

k@t.��x/�1u.t/kL6

b.�/�

1C t

te�˛tQ.ku0kˆb/CQ

�kgkL6

b.�/

�and, therefore, similar to the Corollary 3.3,

kf .u.t//kL6b.�/Cku.t/kW2,6

b .�/Ckrx u.t/kL1.�/ �

1C t

te�˛tQ.ku0kˆb/CQ

�kgkL6

b.�/

�,

where we have implicitly used the maximal regularity theorem for the Laplacian in L6b and the embedding W1,6 � L1. Finally, using this

estimate together with assumptions 4.14, we obtain

kf .u.t//kW1,15=4

b .�/� CkrxukL1.�/kf 0.u.t//k

L15=4b .�/

�

� CkrxukL1.�/

�1Ckf .u.t//kL6

b.�/

�8=5�

1C t13=5

t13=5e�˛tQ.ku0kˆb/CQ

�kgkL6

b.�/

� (4.26)18

96



and the desired estimate (4.25) follows now from the embedding W1,15=4 � L1. �

Remark 4.7Because limu!˙1 f .u/D1, there exists a strictly positive (monotone decreasing) function ıf depending only on f such that

ku.t/kL1.�/ � 1� ıf�kf .u.t//kL1.�/

�. (4.27)

Thus, the estimate (4.26) shows that any solution u.t/ of the problem (2.12) is indeed separated (uniformly in time) from the singularitiesuD˙1. This allows to obtain the further regularity of the solution exactly as in the regular case.

5. Attractors: the case with uniqueness

This section is devoted to the long-time behavior of solutions of the CH problem in the case where the uniqueness theorem holds(more delicate situation where we do not have the uniqueness result will be considered in the next section). We also recall that asusual for the case of unbounded domains (see [21] and references therein), the corresponding attractor is not compact in the (uniform)topology of the initial phase space and does not attract in that space, so one should consider the so-called locally-compact attractors,which attract bounded sets of the initial phase space in the appropriate local topology. For the convenience of the reader, we recall thedefinition of such an attractor adapted to the case of CH equation.

Definition 5.1Let the assumptions of Theorems 4.1 and 4.4 hold and let S.t/ : ˆb ! ˆb be the solution semigroup associated with the CHequation (2.12). A set A is called a locally-compact attractor of the CH equation if the following conditions are satisfied:

(1) A is bounded inˆb and is compact inˆloc :DW1,2loc .�/;

(2) it is strictly invariant: S.t/ADA for t � 0; and(3) it attracts bounded sets ofˆb in the topology ofˆloc , that is, for every B bounded inˆb and every neighborhood O.A/ of the set

A, there exists a time T D T.B,O/ such that

S.t/B�O.A/ if t � T .

Recall that the compactness in ˆloc simply means that the restriction Aˇ̌̌�ŒS1,S2�

to any bounded subcylinder �ŒS1,S2� is compact in

W1,2��ŒS1,S2�

�and the attraction property means that

limt!1

distW1,2.�ŒS1,S2�/

�S.t/B

ˇ̌̌�ŒS1,S2�

,Aˇ̌̌�ŒS1,S2�

�D 0

for any bounded set B ofˆb and any bounded subcyliner�ŒS1,S2�. Here and in what follows, distV denotes the Hausdorff semidistancein V .

Theorem 5.2Let the assumptions of Theorem 4.1 or 4.4 hold. Then, the CH equation possesses a locally-compact attractor A in the sense of theprevious definition, which is generated by all bounded solutions of that equation defined for all t 2R:

ADK jtD0 , (5.1)

where K D fu 2 L1.R,ˆb/, u solves (2.12)g and u jtD0 :D u.0/ . Moreover, this attractor is bounded in W2,2b .�/, and in the singular

case, the attractor is separated from the singularities:

ku0kCb.�/ � 1� ı, 8u0 2A, (5.2)

for some positive ı.

ProofIndeed, thanks to the abstract theorem on the existence of an attractor, we need to construct a bounded in ˆb and (pre)compact inˆloc absorbing set B of the semigroup S.t/ and to verify that S.t/ restricted to B has a closed graph [42].

Assume first that the equation is regular and assumptions of Theorem 4.1 hold. Then, because of Corollary 4.3, the H2b-ball

B :Dn

u 2W2,2b .�/, kukW2,2

b� R

o

is an absorbing set for the semigroup S.t/ (if R is large enough), which is obviously compact in ˆloc . The fact that the graph of S.t/ isclosed on B is an immediate corollary of the Lipschitz continuity (4.2) in a weaker topology.


18

97


Let now the equation be singular and the assumptions of Theorem 4.4 hold. Then, because of Corollary 4.5, Proposition 4.6 andestimate (4.27), the set

B :Dn

u 2W2,2b .�/, kukW2,2

b� R, kukCb.�/ � 1� ı

o

will be an absorbing set for the semigroup S.t/ (if R is large enough and ı > 0 is small enough), and it is again compact inˆloc . The factthat the graph of S.t/ is closed is again an immediate corollary of the Lipschitz continuity (4.2) proved in Theorem 4.4.

Thus, the assumptions of the abstract attractors existence theorem are verified in both cases. The description (5.1) is also a standardcorollary of that theorem, and the fact that attractor is bounded in W2,2

b and in the singular case is separated from singularities is alsoimmediate because the attractor is a subset of any absorbing set. Theorem 5.2 is proved. �

Remark 5.3The regularity of the obtained attractor may be further improved. Indeed, because the solution on the attractor is proved to be sepa-rated from singularities and globally bounded, the usual bootstrapping arguments show that the factual regularity of the attractor isrestricted only by the smoothness of the domain �, nonlinearity f and the external forces g (if all of them are C1 regular, then so willbe the attractor).

We conclude the section by a brief discussion of the further qualitative properties of the constructed global attractor. Note that theattractors in unbounded domains usually have very complicated structure whose complete description is not available even in the sim-plest case of real Ginzburg–Landau equation. However, a number of important and interesting features can be established followingthe general strategy of investigating the dissipative PDEs in unbounded domains, see [21] for the detailed exposition. Although manyof the results, which will be stated, can be treated as formally new, they can be obtained as a straightforward application of generaltechnique to the concrete case of the CH equation (as far as its well-posedness in the uniformly local spaces is established). By thisreason, we restrict ourselves just to announcing these results and will return to the detailed proofs somewhere else.

We start with the dimension. As usual, the infinite-dimensionality of the attractor can be established using the essentially unstablemanifolds. We remind that the essentially unstable manifold of an equilibrium z0 is generated by all trajectories u.t/, t 2 R, whichconverge ‘fast’ to z0 as t!�1, namely,

ku.t/� z0kL2b� Ce�t , t � 0

with �� 0 > 0 and �0 is determined by the spectral data of the equilibrium z0, see [21].

Corollary 5.4Let the assumptions of Theorem 5.2 hold and let, in addition, f .0/D 0, g� 0 and the zero equilibrium of (2.12) exponentially unstable,that is, f 0.0/ < ��1 where �1 > 0 is the first eigenvalue of the Laplacian on a cross-section ! with Dirichlet boundary conditions. Then,the attractor A contains an infinite-dimensional submanifold – the essentially unstable manifold of zero equilibrium and, therefore, itsHausdorff and fractal dimension is infinite.

The proof of this theorem is a straightforward application of the essentially unstable manifold theorem for maps stated in [43] and,by this reason, is omitted, see also [21, 29, 31, 40] for applications of that theorem for various equations in unbounded domains.

The ‘thickness’ of the infinite-dimensional attractor as well as some of topological features of the dynamics on it can be describedusing the so-called Kolmogorov’s "-entropy. Actually, the sharp upper and lower bounds for that entropy can be obtained follow-ing the aforementioned general scheme. We will not give more details here sending the interested reader to the survey [21] and tothe references therein. Instead, we mention in what follows the particular class of nonlinearities f and external forces g for which thefinite-dimensionality of the global attractor can be established.

Corollary 5.5Let the assumptions of Theorem 5.2 hold and let, in addition, assumption (2.21) hold, and g.x/ tends to zero as jxj !1 in the followingsense:

kgkL2.�Œs,sC1�/! 0

as jsj ! 1. Then the attractor A has finite fractal dimension. Moreover, the attractor A is exponentially close to any equilibriumu0 D u0.x/ of the CH problem as jxj !1, that is,

ju.x/� u0.x/j � Ce�˛jx1j, x 2�, u 2A,

where the positive constants C and ˛ are independent of x and u 2A.

The proof of this corollary is a straightforward adaptation of [30] (see also [21]) to the case of CH equations and, by this reason, is alsoomitted.

Finally, we discuss one more interesting topic – the existence and stability of the so-called kink solutions, which play an essential rolein the study of coarsening dynamics in phase transitions.1

89

8



Corollary 5.6Let the assumptions of Corollary 5.4 hold and let, in addition, the nonlinearity f be odd: f .�u/D�f .u/. Then there exists a positive equi-librium u0.x0/ � 0, x0 2 ! of the CH problem depending only on the cross-section variable x0 and a monotone kink-type equilibriumuk.x/ connecting u0.x0/ and�u0.x0/ as x1!˙1, that is,

limx1!˙1

supx12!juk.x1, x0/� u0.x

0/j D 0, @x1 uk.x/� 0. (5.3)

ProofActually, there is a vast amount of literature devoted to kink-type solutions of reaction–diffusion equations (see [44] and referencestherein). Nevertheless, we failed to find a sharp reference, so we sketched in what follows an elementary proof of the existence result.

The desired kink solution uk can be constructed as a limit as t!1 of the associated second-order parabolic equation:

@tuD�xu� f .u/, uˇ̌̌RC�@! D 0 (5.4)

in a half cylinder �C :D RC � ! with the extra boundary condition u jx1D0 D 0 (the part of the kink, which corresponds to x1 < 0,is obtained after that by reflection). Indeed, because the zero equilibrium is unstable, one can easily construct a smooth boundedfunction h.x/, x 2�C such that

@x1 h� 0, hˇ̌̌@�C

D 0, �xh� f .h/� 0

(actually, h can be found in the form ".x1/e.x0/ where e is the first eigenvector of the Laplacian in ! and ".x1/ is a small monotone-increasing function). Then, solving (5.4) with u.0/ D h, we see that the solution u.t, x/ will be bounded and increasing in t for everyfixed x and in x1 for every fixed t and x0, because of the maximum principle. Thus, uk.x/ :D limt!1 u.t, x/ exists and gives us the desiredkink solution. �

The stability analysis of such kinks is much simpler than in the case of the whole space (see, e.g., [23, 24]). Indeed, under the genericassumption that the limit state u0.x0/ is non-degenerate, the essential spectrum of the kink will be separated from zero, so the nonlinearstability can be derived from the linear one in a straightforward way using the center manifold technique. The linear stability analysisis also straightforward because of the following elementary observation: the linear stability of the kink uk.x/ for the case of reaction-diffusion problem (5.4) in � implies its linear stability for the case of full CH equation. But (5.4) is an order-preserving system and thelinear stability follows from monotonicity because of the Perron–Frobenius theory, see for example, [45]. Combining these arguments,we end up with the following result.

Corollary 5.7Let the assumptions of Corollary 5.6 hold and let, in addition, the limit state u0.x0/be non-degenerate. Then, the associated kink solutionuk.x/ is exponentially stable, that is, there exists " > 0 such that, for any w 2 Cb.�/ such that

kw � ukkCb � ",

there exists a shifted kink uk,� .x/ :D uk.x1 � � , x0/, � 2R, such that

kS.t/w � uk,�kCb � Ce�˛t ,

where S.t/ is an evolutionary semigroup generated by the CH equation and C and ˛ are the appropriate positive constants.

Remark 5.8Of course, the kink solution uk as well as all its shifts uk,� belong to the attractor A (just stationary points). In addition, using the centermanifold approach, one may study the coarsening dynamics and slow relaxation of the multi-kink structures (spatially well-separatedpairs of kink–antikink solutions) and write out the system of ODEs for the temporal evolution of the centers of that kinks, analogous to[46] (see also [47] for more general construction). Most part of such solutions on the center manifold can be extended for all negativetimes and, therefore, also belong to the attractor. We will return to this subject somewhere else.

6. Attractors: the case without uniquenessWe now turn to discuss the case where the uniqueness theorem does not hold and only the assumptions of Theorem 3.1 are satisfied,in particular, it will be so for the case of logarithmic potential (4.15). Because we only have the existence (but not uniqueness) of asolution, we will use the so-called trajectory approach, see [21, 48] for more details.

As a first step, following the general scheme, we introduce the so-called trajectory dynamical system associated with the CHequation (2.12).

Definition 6.1Let a set KC � L1.RC,ˆb/ be the set of all weak solutions u : RC ! ˆb of the CH problem (2.12) in the sense of Definition 2.6(because the �-component of the solution is uniquely determined by its u-component, we omit the �-part of the solution in thatdefinition) such that the assumptions (2.19) are satisfied and the following analog of the main dissipative estimate holds:


18

99


ku.t/k2W1,2

b .�/CkF.u.t//kL1

b.�/Ck�xuk2

L2b.Œt,tC1��/

C

Ckf .u/k2L2

b.Œt,tC1��/Ckrx�k

2L2

b.Œt,tC1��/� Cue�˛t C C

�kgk2

L2b.�/C 1

�,

(6.1)

where the constant C is the same as in estimate (3.1), t � 0 is arbitrary and Cu is some positive number depending on the trajectory u.It is not difficult to verify that the translation semigroup

T.h/ : KC! KC, .T.h/u/.t/ :D u.tC h/ (6.2)

acts on the set KC and, in the case with uniqueness, this semigroup is conjugated to the standard solution semigroup S.h/ acting on theusual phase space ˆb. By this reason, the set KC is called the trajectory phase space of problem (2.12) and the translation semigroupT.h/ acting on this space is often referred as a trajectory dynamical system associated with this equation.

We intend to find an attractor for the introduced trajectory dynamical system (trajectory attractor for the initial CH equation). To thisend, as usual, we need to specify the class of ‘bounded’ sets in KC and fix the appropriate topology in KC.

Definition 6.2A set of trajectories B� KC is bounded if the dissipative estimate (6.1) holds uniformly with respect to all u 2 B with the same constantCB, that is,

Cu � CB <1, 8u 2 B.

Comparing the dissipative estimates (3.1) and (6.1), we see that at least in the case of uniqueness, the constant Cu is simply related withtheˆb-norm of the initial data u0. Therefore, in the case of uniqueness, the class of bounded sets thus defined corresponds to the usualbounded sets in the phase space B and gives a natural extension of that concept to the case without uniqueness.

In this section, we will consider only the so-called weak (trajectory) attractors, so we introduce the topology in KC in the followingway.

Definition 6.3We endow the trajectory phase space KC by the weak-star topology of the space

‚C :D L1loc

�RC, W1,2

loc .�/�\ L2

loc

�RC, W2,2

loc .�/�

.

We recall that un ! u in that topology iff, for every time segment Œt, t C T� and every finite cylinder �Œ�S,S�, un ! u weak-star in

L1�Œt, tC T�, W1,2.�Œ�S,S�/

�and weakly in L2

�Œt, tC T�, W2,2.�Œ�S,S�/

�.

It is important for the attractor theory that every bounded set of ‚C is precompact and metrizable in this weak-star topology(see [48, 49] for the details).

We are now ready to define attractor of the trajectory semigroup T.h/ acting on KC.

Definition 6.4A set Atr � KC is a global attractor of the trajectory dynamical system .T.h/, KC/ (trajectory attractor of the CH equation (2.12)) if

(1) Atr is compact in KC (endowed by the weak-star topology of KC);(2) strictly invariant: T.h/Atr DAtr ; and(3) attracts all bounded (in the sense of Definition 6.2) sets of KC in the weak-star topology of‚C.

Finally, the next theorem gives the existence of such an attractor.

Theorem 6.5Let the assumptions of Theorem 3.1 hold. Then, the CH equation (2.12) possesses a trajectory attractor Atr in the sense of the previousdefinition. Moreover, this attractor is also generated by all bounded solutions of problem (2.12) defined for all t 2R:

Atr DK jt�0 , (6.3)

where K� L1.R,ˆb/ is a set of all solutions u : R!ˆb, which are defined for all t and satisfy the dissipative estimate (6.1) for all t 2Rwith Cu D 0.

ProofWe first note that the set KC is not empty because of Theorem 3.1, so the trajectory dynamical system is reasonably defined. Next,analyzing the dissipative estimate (6.1) and the definition of a bounded set, we see that the set B � KC of all trajectories u, whichsatisfy this estimate with the constant Cu � 1, is a bounded absorbing set of the semigroup T.h/ and even

T.h/B � B

19

00



for all h 2 RC. Moreover, B is precompact and metrizable in the topology of KC (weak-star topology of ‚C) and clearly the shiftsemigroup T.h/ is continuous in that topology. Thus, in order to be able to apply the appropriate abstract attractor existence theorem(see [21], Theorem 2.20; see also [42,48] and references therein), we only need to verify that B is closed as a subset of‚C with the weak-star topology. Because this proof repeats word by word the proof of the existence of a weak solution (which is constructed exactly bythe weak-star limit of solutions of the appropriate approximate problem), we rest it for the reader.

Thus, because of the abstract attractor existence theorem, the trajectory attractor exists and is generated by all bounded completesolutions of the CH problem (2.12). Theorem 6.5 is proved. �

Remark 6.6Using the fact that the appropriate norm of the time derivative @tu is under the control for every weak solution u (because of the dis-sipative estimate and the first equation (2.12)), one can verify that the trajectory attractor Atr attracts bounded sets of KC in a strongtopology of

‚C. / :D L1loc

�RC, W1�,2

loc .�/�\ L2

loc

�RC, W2�,2

loc .�/�

,

for every > 0. The strong attraction in the space‚C. /with D 0 is more delicate and will be proved in Section 8 using the so-calledweighted energy method.

7. Weighted energy equalities

In this section, we will mainly consider the case of singular potentials without uniqueness. We first check that any weak solution of theCH satisfies the weighted energy equalities and then, in the next section, prove that the weak trajectory attractor constructed before iscompact in a strong topology and the attraction holds in the strong topology as well.

We start with the following lemma, which is the key part of our proof of the weighted energy equalities.

Lemma 7.1Let the function u : RC ��!R be such that8<

:u 2 L1

�RC, W1,2

b .�/�\ L2

b

�RC, W2,2

b .�/�

,

@tu 2 L2b

�RC, W�1,2

b .�/�

, F.u/ 2 L1�RC, L1

b.�/� (7.1)

and

H :D�xu� f .u/ 2 L2b

�RC, W1,2

b .�/�

, (7.2)

where the nonlinearity f satisfies assumptions (2.14). Then, for all weight functions ' 2 L1.R/ satisfying (2.7) (with some positive ") andalmost all T1, T2 2RC, T2 > T1, we have

E'.u.T2//� E'.u.T1//D�

Z T2

T1

.'0@x1 u.t/, @tu.t//� .H.t/,'@tu.t//dt, (7.3)

where

E'.u/ :D1

2

�', jrxuj2

�C .'F.u/, 1/. (7.4)

Remark 7.2The main difficulty in the proof of this and the next lemmas is that we do not have the maximal W1,2 ! W3,2-regularity forequation (7.2) treated as a semilinear elliptic equation

�xu� f .u/D H.

By this reason, we are unable to deduce that�xu and f .u/ separately belong to W1,2b . Thus, although the inner product .'@tu,�xu�f .u//

is well-posed, the terms .'�xu, @tu/ and .'@tu, f .u// can be nevertheless ill-posed and we cannot use the standard methods to verifythe energy equality. Instead of that, we obtain the result (following [35]) by using the trick based on the convexity arguments. Analternative method, based on the abstract energy equality for the maximal monotone operators [50], can be found in [51] Lemma 4.1(the analogous result is proved there for the case of bounded domains).

Proof of the lemmaWe first note that without loss of generality, we may think that the potential F is convex. Indeed, it follows from the conditions (2.14) thatthere exists K > 0 such that f0.u/ :D f .u/C Ku is monotone and, therefore, the associated potential F0.u/ :D

R u0 f .v/dv D F.u/C Ku2=2

is convex. On the other hand, the regularity (7.1) is enough for verifying that

1

2

d

dt

�u2,'

�D .@tu,'u/


19

01


in a straightforward way. Thus, the general case differs from the convex one by the presence of the linear term f0.u/� f .u/D Ku in (7.2),which can be easily treated. By this reason, we consider in what follows only the case when the potential F is convex.

Then we may write out the following inequalities, which hold for all h > 0:

F.u. C h//D F.u.//C F0.u.//.u. C h/� u.//

C1

2ŠF00.�u. C h/C .1� �/u.//.u. C h/� u.//2, � 2 .0, 1/.

Because F./ is convex, we have

F.u. C h//� F.u.//� F0.u.//.u. C h/� u.//

that is,

1

hŒF.u. C h//� F.u.//��

1

hf .u.// Œu. C h/� u./� . (7.5)

Similarly, we obtain

F.u.//D F.u. C h//C F0.u. C h//.u./� u. C h//

C1

2ŠF00.�u. C h/C .1� �/u.//.u. C h/� u.//2, � 2 .0, 1/.

Because F00./� 0, we infer from the last inequality that

1

hŒF.u. C h//� F.u.//��

1

hf .u. C h// Œu. C h/� u./� . (7.6)

Multiplying these inequalities by ' and integrating over x, we end up with

�f .u.//,'

u. C h/� u./

h

��

�',

F.u. C h//� F.u.//

h

��

�f .u. C h//,'

u. C h/� u./

h

�(7.7)

(actually, because of our assumptions on u, all terms in that inequality are well defined for almost all ). In addition, because.', jrxu./j2/ is also a convex functional, we have the analogous, but simpler inequalities

�rxu./,'

rx u. C h/�rxu./

h

��

1

2

�',jrxu. C h/j2 � jrxu./j2

h

��

�

�rxu. C h/,'

rxu. C h/�rxu./

h

�.

(7.8)

Taking a sum of these two inequalities, integrating by parts and using the definition of H./, we arrive at

��H./,'

rxu. C h/�rxu./

h

��

�@x1 u./,'0

u. C h/� u./

h

��

E'.u. C h//� E'.u.//h

�

��H. C h/,'

rxu. C h/�rxu./

h

��

�@x1 u. C h/,'0

u. C h/� u./

h

�.

(7.9)

Finally, integrating this formula over 2 ŒT1, T2�, we have

Z T2

T1

��H./,'

rxu. C h/�rxu./

h

��

�@x1 u./,'0

u. C h/� u./

h

�

d �1

h

Z T2Ch

T2

E'.u.//d �1

h

Z T1Ch

T1

E'.u.//d �

Z T2

T1

��H. C h/,'

rxu. C h/�rxu./

h

�

�

�@x1 u. C h/,'0

u. C h/� u./

h

�d .

(7.10)

It only remains to note that, because of our assumptions on u, we may pass to the limit h! 0 for almost all fixed T1 and T2. This givesthe desired equality (7.3). �

As the next step, we need (7.3) to hold for every T1, T2 2RC. That is proved in the following lemma.

19

02



Lemma 7.3Let the assumptions of Lemma 7.1 hold and let, in addition, the function u 2 C.Œ0, T�, L2

'.�// and the function ! E'.u.// be lowersemicontinuous, that is,

E'.u.//� lim infn!1

E'.u.n//

for any n! and any . Then, ! E'.u.// is absolutely continuous and (7.3) holds for all T1 > 0 and T2 > 0.

ProofWe first note that because of the lower semicontinuity,

E'.u.T2//� E'.u.T1//� �

Z T2

T1

.@x1 u.t/,�[email protected]//� .H.t/, @tu.t//dt (7.11)

for all T2 and almost all T1. Assume now that we proved that this inequality holds for all T1 as well. Then the assertion of the lemmaholds. Indeed, we have the strict inequality for some T1 > 0 and T2 > 0. Then, we may find T�1 < T1 and T�2 > T2 such that the equalityholds on the interval ŒT�1 , T�2 � (because it holds for almost all T1 and T2). Splitting the interval ŒT�1 , T�2 �D ŒT

�1 , T1�[ ŒT1, T2�[ ŒT2, T�2 � and

using (7.11) for first and third interval together with the strict inequality on the second interval, we see that the inequality must be strictalso on the interval ŒT�1 , T�2 �. Thus, in order to prove the lemma, we only need to verify that inequality (7.11) holds for every T1. Withoutloss of generality, we may prove that for T1 D 0 only.

To this end, we note that the function u is a unique solution of the following CH type problem

'�1=2.��x/�1�'1=2@tv

�D '�1rx.'rxv/� f .v/� Qg.t/, v jtD0 D ujtD0 (7.12)

with Qg.t/ :D H.t/� '�1=2.��x/�1.'1=2@tu/C '�1'0@x1 u. Indeed, by the construction u is a weak solutions of that equation. Let v1.t/

and v2.t/. Then, writing out the equation for the difference wD v1�v2, multiplying this equation by 'w and arguing in a standard way(exploiting the monotonicity of f ), we arrive at

kw.t/kW�1,2'� CeKtkw.0/kW�1,2

'

and the uniqueness holds.Note also that H 2 L2

b.RC, W1,2b .�// by the assumptions on u. So, we only need to verify the energy inequality for the auxiliary

equation (7.12). To this end, we approximate the singular potential f by the regular ones fn just by replacing f .u/ outside of.�1C 1=n, 1� 1=n/, by the proper linear function:

'�1=2.��x/�1�'1=2@tvn

�D '�1rx.'rxvn/� fn.vn/� Qg.t/, vn jtD0 D ujtD0 . (7.13)

Then, because (7.13) is a small compact perturbation of the standard CH equation in � with linearly growing nonlinearity fn, we havethe unique solvability as well as the energy equality:

E',n.vn.T//� E',n.u.0//C

Z T

0k'[email protected]/k

2W�1,2 dtD

Z T

0.' Qg.t/, @tvn.t//dt.

where E',n.z/ is the energy with the potential F replaced by Fn. Now, passing to the limit n ! 1, vn ! u and using the obviousrelations

E',n.u.0//! E'.u.0//; E'.u.T//� lim infn!1

E',n.vn.T//

together with the weak convergence @tvn! @tu in L2.0, T , W�1,2' .�// (these results are standard for the CH equations theory, so we do

not present the proofs here, see [17] for more details), we arrive at

E'.u.T//� E'.u.0//CZ T

0k'[email protected]/k2

W�1,2 dtD

Z T

0.' Qg.t/, @tu.t//dt

which is equivalent to the desired energy inequality (7.11) and that finishes the proof of the lemma. �

Remark 7.4Being pedantic, we have proved in Lemma 7.3 that the energy equality holds for T1 > 0 only. This drawback can be easily corrected bythe proper extending of the function u.t/ to negative times and arguing as before. Thus, (7.3) remains true for T1 D 0 as well.


19

03


We are now able to return to the initial CH problem.

Corollary 7.5Let u be a weak solution of the CH problem (2.12). Then, for any weight function ' 2 L1 satisfying (2.7), the function t ! E'.u.t// isabsolutely continuous on Œ0,1/ and the following energy identity holds:

d

dt

�E'.u.t//C .u', g/

�C�', jrx�.t/j

2�C [email protected]/,'0@x1 u.t//C .@x1�.t/,'

0�.t//D 0 (7.14)

for almost all t.

Indeed, in order to obtain (7.14), it is sufficient to multiply the initial equation (1.1) by '� and use the energy equality proved inLemma 7.3 (all the assumptions of the lemma are clearly satisfied for any weak solution).

Corollary 7.6Let u be a weak solution of the CH problem (2.12) in the case of singular potential f . Then, u 2 C.Œ0, T�, W1,2

' .�// and F.u/ 2C.Œ0, T�, L1

'.�//, for any T > 0 and any integrable weight ' satisfying (2.7) (with some positive ").

ProofWithout loss of generality, we prove the continuity at t D 0 only. Indeed, let tn ! 0. Then, from the energy equality, we see thatE'.u.tn//! E'.u.0//. Then, from the convexity arguments and using that u 2 C.Œ0, T�, L2

'/, we see that

ku.tn/kL2'!ku.0/kL2

', kF.u.tn//kL1

'!kF.u.0//kL1

'.

The first convergence together with the obvious weak convergence u.tn/! u.0/ in W1,2' gives the strong convergence in that space.

Analogously, the second convergence together with the convergence F.u.tn// ! F.u.0// almost everywhere and the standard factfrom the Lebesgue integration theory (let zn � 0, zn! z almost everywhere and

Rzn!

Rz, then zn! z in L1) imply the convergence

F.u.tn//! F.u.0// in L1'.�/, and the corollary is proved. �

We conclude this section by stating one more straightforward observation, which is however important for what follows.

Proposition 7.7

Let the function u 2 W2,2b .�/ \ fu

ˇ̌@�D 0g be such that f .u/ 2 L2

b.�/ and let F1=2.u/ :DR u

0

qf 00.v/dv (2.16) and ' be the integrable

weight satisfying (2.7). Then, F1=2.u/ 2W1,2b .�/ and

.rx.'rxu/, f0.u//D��', jrx F1=2.u/j

2�

. (7.15)

ProofIndeed, approximating the singular function f0.u/ by the regular ones fn.u/ (just replacing f 00.u/ by the appropriate constant when u isoutside of .�1C 1=n, 1� 1=n/, we see that clearly, fn.u/! f0.u/ almost everywhere and in L2

'.�/. On the other hand, because fn.u/ is

now a regular C1-function,

.rx.'rxu/, fn.u//D��', f 0n.u/jrxuj2

�.

Passing to the limit here and using that f 0n is monotone-increasing, we conclude that f 00.u/jrxuj2 2 L1'.�/ and

.rx.'rxu/, f0.u//D��', f 00.u/jrxuj2

�.

Thus, we only need to prove that rxF1=2.u/ Dq

f 00.u/rxu in the sense of distributions, but that can be easily verified, for example,

by truncating the function u with an appropriate constant outside of .�1 C 1=n, 1 � 1=n/ and passing to the limit n ! 1). Thus,Proposition 7.7 is proved. �

Corollary 7.8Any weak energy solution u of the CH problem (2.12) satisfies

F1=2.u/ 2 L2�Œ0, T�, W1,2

' .�/�

,

for every T and every integrable weight '.

Indeed, we have the control of the L2' -norm of the solution u from the energy estimate. Thus, the assertion is an immediate corollary

of the previous proposition.

19

04



8. Strong attraction via the energy method

In this concluding section, we develop the weighted energy method and improve essentially the results on the trajectory attractorsobtained in Section 6. We start with simplifying the construction of the trajectory dynamical system (Definitions 6.1 and 6.2).

Proposition 8.1Under the assumptions of Theorem 6.5, any weak solution u of the CH problem (Definition 2.6) satisfies estimates (3.1) and (6.1) withCu :D Cku.0/k2

ˆb. Thus, condition (6.1) in the Definition 6.1 is automatically satisfied (and can be omitted). Moreover, the class of

bounded sets introduced in Definition 6.2 possesses the following natural description:

B� KC is bounded if and only if B jtD0 :D fu.0/, u 2 Bg is bounded inˆb.

Indeed, because the weighted energy equality holds for every weak solution u, repeating word by word the derivation of (3.1), wederive estimate (6.1) for any weak solution u with Cu :D Cku.0/k2

ˆb. The other assertions of the proposition follow immediately from

this estimate.We now introduce the natural strong topology on the space KC.

Definition 8.2Let F1=2.u/ :D

R u0

qf 00.v/dv and let

‚Cstrong :Dn

u 2 Cloc

�RC, W1,2

loc .�/\ L2loc

�RC, W1,2

loc .�/�\W1,2

loc

�RC, W�1,2

loc .�/�

,

�� 2 L2loc

�RC, W1,2

loc .�/�

, F.u/ 2 Cloc

�RC, L1

loc.�/�

,

�f .u/ 2 L2loc.RC ��/, F1=2.u/ 2 L2

loc

�RC, W1,2

loc .�/�o

.

(8.1)

Then the topology induced by the embedding KC �‚Cstrong is called a strong topology on the trajectory phase space KC.

Remark 8.3The previous definition is not self-contradictory, because any weak solution u belongs to ‚Cstrong and KC is a subset of that space,see Corollaries 7.6 and 7.8.

We are now able to state the main result of this section.

Theorem 8.4Let the assumptions of Theorem 6.5 hold and let f .u/ be a singular potential. Then, the trajectory dynamical system .T.h/, KC/ associ-ated with the CH equation possesses the compact global attractor in the strong topology of‚Cstrong. Moreover, this attractor coincideswith the weak trajectory attractor constructed in Theorem 6.5.

ProofIndeed, let hn !1, un 2 KC from a bounded set (so kun.0/kˆb � C) and vn :D T.hn/un. Then, according to Theorem 6.5, without lossof generality, we may assume that vn ! u in a weak topology of‚C. And to prove the theorem, we only need to verify that vn ! u ina strong topology.

Let ' be an integrable weight satisfying (2.7), and let us rewrite the energy equality (7.14) for vn as follows:

E'.vn.T//C .vn.T/', g/� ŒE'.un.0//C .un.0/', g/�e�˛.TChn/C

C

Z T

�hn

e�˛.T�t/H'.vn.t//dtD 0(8.2)

with

H'.vn.t// :D.', jrx�n.t/j2/C [email protected]/,'

0@x1 vn.t//C

C .@x1�n.t/,'0�n.t//� ˛ŒE'.vn.t//C .vn.t/', g/�

(8.3)

and with sufficiently small ˛ > 0, which will be specified later, and any T > 0. And of course, for the limit function u.t/, we have thefollowing energy equality:

ŒE'.u.T//C .u.T/', g/�C

Z T

�1e�˛.T�t/H'.u.t//dtD 0. (8.4)

As usual for the energy method, we need to pass to the weak limit n!1 in (8.2) and compare the result with the limit equation (8.4).Indeed, for the first term, we have as before,

E'.u.T//� lim infn!1

E'.vn.T//


19

05


and .vn.T/', g/! .u.T/', g/. The second term obviously tends to zero (because un.0/ are bounded and hn!1), so, we only need toestablish the analogous inequality for the third term. To this end, we transform it as follows:

H'.vn/D1

4

�', jrx�n.t/j

2�C

1

4

�', jrx�n.t/j

2�� 2ˇ

�', j�n.t/j

2�C

C ˇh�', j�xvn.t/j

2�C�', jf .vn/j

2�C 4

�', jrxF1=2.vn.t//j

2�iC

Ch

4ˇ.'0@x1 vn.t/, f .vn.t///� 4ˇK�', jrxvn.t/j

2�� ˛.vn.t/', g/

iC

Ch�',ˇjf .vn.t//j

2 � ˛F.vn.t//�iC

C

ˇ�', j�xvn.t/j

2�C

1

2

�', jrx�n.t/j

2��

1

2˛�', jrxvn.t/j

2�C

C.@x1�n.t/,'0�n.t//C

�.�x/

�1�n.t/,'0@x1 vn.t/

�iD

D I1.vn/C I2.vn/C I3.vn/C I4.vn/C I5.vn/C I6.vn/,

(8.5)

where ˇ > 0 is a sufficiently small positive number. Let us pass to the limit in every term of (8.5) separately. Indeed, without loss ofgenerality, �n! �weakly in L2.ŒS, SC 1�, W1,2

' /, for all S,

Z T

�1e�˛.T�t/I1.u.t//dt � lim inf

n!1

Z T

hn

e�˛.T�t/I1.vn.t//dt.

The analogous estimate for the second term follows from the fact that (because of the Friedrichs inequality) this term is a posi-tive definite quadratic form. The estimate for the third term follows from the weak convergences �xvn ! �xu, f .vn/ ! f .u/ andrxF1=2.vn/!rxF1=2.u/ in the space L2.ŒS, SC 1�, L2

'.�//.

The fourth term is trivial because we have the strong convergence rxvn !rx u in L2.ŒS, SC 1�, L2'.�//. The desired estimate for the

fifth term follows from the convergence vn! u almost everywhere, the estimate ˇf 2.z/�˛F.z/� �C and the Fatou lemma. Finally, thesixth term is a quadratic form with respect to vn and�n, which will be also positive definite if " > 0 (from (2.7)) and ˛ are small enough.Thus, the desired estimate for I6 is also true, and we arrive at

Z T

�1e�˛.T�t/H'.vn.t//dt � lim inf

n!1

Z T

hn

e�˛.T�t/H'.u.t//dt.

Comparing this estimate with (8.4), we see that it is possible only if all of the previous inequalities are, in fact, equalities. Thus, we havefactually verified that

k�nkL2.ŒS,SC1�,W1,2' /!k�kL2.ŒS,SC1�,W1,2

' /,

k�xvnkL2.ŒS,SC1�,L2'/!k�xukL2.ŒS,SC1�,L2

'/,

krxF1=2.vn/kL2.ŒS,SC1�,L2'/!krxF1=2.u/kL2.ŒS,SC1�,L2

'/,

kf .vn/kL2.ŒS,SC1�,L2'/!kf .u/kL2.ŒS,SC1�,L2

'/,

krxvn.T/kL2'!krx u.T/kL2

',

kF.vn.T/kL1'!kF.u.T//kL1

'

(8.6)

for all fixed S and T . That, together with the weak convergences, implies the desired strong convergence for �n, �xvn, rx F1=2.vn/ andf .vn/. Thus, we only need to verify that

vn! u in C�ŒS, SC 1�, W1,2

'

�and F.vn/! F.u/ in C

�ŒS, SC 1�, L1

'

�.

We note that, although the last two convergences of (8.6) imply that vn.T/ ! u.T/ in ˆ' , that does not give straightforwardly thedesired uniform convergence in C.ŒS, SC 1�,ˆ'/. However, these convergences imply in a standard way that the trace A :D AtrjtD0 ofthe trajectory attractor Atr to the phase space is compact inˆ' and that

sups2ŒS,SC1�

dˆ' .vn.s/,A/! 0 (8.7)

as n!1. Let us show that it is sufficient to verify the desired uniform convergences. Indeed, let us fix the orthonormal basis en in thespace W1,2

' and the corresponding orhtoprojector PN of the first N vectors and QN :D 1� PN. Then, on the one hand,

PNvn! PNu

19

06



in C.ŒS, SC1�, W1,2' / for any N (because we have from the very beginning the strong convergence in C.ŒS, SC1�, L2

'/. On the other hand,

(8.7) together with the compactness of the attractor A in W1,2' implies that for any " > 0, we may find ND N."/ such that

lim supn!1

kQNvnkC�ŒS,SC1�,W1,2

'

� � ".

Thus, the uniform convergence in C.ŒS, SC 1�, W1,2' / is verified.

Let us now prove that F.vn/! F.u/ in C.ŒS, SC 1�, L1'/. To this end, we note that, because of the absolute continuity of the Lebesgue

integration, the compactness of F.A/ in L1' and convergence (8.7) imply that for every " > 0, there is ı D ı."/ > 0 such that

lim infn!1

sups2ŒS,SC1�

ZA

F.vn.s//dx � "

if mes'.A/ :DR

A ' dx < ı. The desired uniform convergence F.vn/ ! F.u/ in C.ŒS, S C 1�, L1'/ is now an immediate corollary of the

Egorov theorem. Thus, all desired strong convergences are verified and the theorem is proved. �

Acknowledgements

The work is supported by TUBITAK, ISBAB project No:107T896 and Russian Government Grant No: 8502. The Authors would like to thankGiulio Schimperna for fruitful discussions.

References1. Bai F, Elliott C, Gardiner A, Spence A, Stuart A. The viscous Cahn-Hilliard equation. Part I: computations. Nonlinearity 1995; 8:131–160. DOI:

10.1088/0951-7715/8/2/002.2. Elliott C. The Cahn-Hilliard Model for the Kinetics of Phase Separation. Mathematical Models for Phase Change Problems, Vol. 88. International Series of

Numerical Mathematics: Birkhauser, Basel, 1989.3. Elliott C, Stuart A. Viscous Cahn-Hilliard equation. II. Analysis. Journal for Differential Equations 1996; 128(2):387–414. DOI: 10.1006/jdeq.1996.0101.4. Kalantarov VK. Global behavior of the solutions of some fourth-order nonlinear equations. (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst.

Steklov. (LOMI) 1987, Kraev. Zadachi Mat. Fiz. i Smezhn. Vopr. Teor. Funktsii, 19; 163:66–75.5. Cahn JW, Hilliard JE. Free energy of a nonuniform system. I. Interfacial free energy. Journal of Chemical Physics 1958; 28:258–267. DOI:

10.1063/1.1744102.6. Debussche A. A singular perturbation of the Cahn-Hilliard equation. Asymptotic Analysis 1991; 4:161–185. DOI: 10.3233/ASY-1991-4202.7. Eden A, Kalantarov VK. The convective Cahn-Hilliard equation. Applied Mathematics Letters 2007; 20:455–461. DOI: 10.1016/j.aml.2006.05.014.8. Efendiev M, Gajewski H, Zelik S. The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity. Advances

in Differential Equations 2002; 7(9):1073–1100.9. Efendiev M, Miranville A, Zelik S. Exponential attractors for a singularly perturbed Cahn-Hilliard system. Math. Nachr. 2004; 272:11–31. DOI:

10.1002/mana.200310186.10. Evans JD, Galaktionov VA, Williams JF. Blow-up and global asymptotics of the limit unstable Cahn-Hilliard equation with a homogeneous

nonlinearity. Siam Journal on Mathematical Analysis 2006; 38:64–102. DOI: 10.1137/S0036141004440289.11. Grasselli M, Schimperna G, Zelik S. On the 2D Cahn-Hilliard equation with inertial term. Communications in Partial Differential Equations 2009;

34(1-3):137–170. DOI: 10.1080/03605300802608247.12. Grasselli M, Schimperna G, Segatti A, Zelik S. On the 3D Cahn-Hilliard equation with inertial term. Journal of Evolution Equations 2009; 9(2):371–404.

DOI: 10.1007/s00028-009-0017-7.13. Grasselli M, Petzeltova H, Schimperna G. Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term. Journal for

Differential Equations 2007; 239(1):38–60. DOI: 10.1016/j.jde.2007.05.003.14. Eden A, Kalantarov VK. 3D Convective Cahn–Hilliard equation. Communications on Pure and Applied Analysis 2007; 6(4):1075–1086. DOI:

10.3934/cpaa.2007.6.1075.15. Miranville A, Zelik S. Doubly nonlinear Cahn-Hilliard-Gurtin equations. Hokkaido Mathematics Journal 2009; 38(2):315–360.16. Miranville A, Zelik AS. Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. Mathematical Methods in the Applied

Sciences 2005; 28(6):709–735. DOI: 10.1002/mma.590.17. Miranville A, Zelik S. Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Mathematical Methods in the Applied

Sciences 2004; 27(5):545–582. DOI: 10.1002/mma.464.18. Novick-Cohen A. The Cahn-Hilliard equation: mathematical and modeling perspectives. Advances in Mathematical Science and Application 1998;

8:965–985.19. Novick-Cohen A. Blow up and growth in the directional solidification of dilute binary alloys. Applied Analysis 1992; 47:241–257. DOI:

10.1080/00036819208840143.20. Winter M. Stationary Solutions For The Cahn-Hilliard Equation. Ann. Inst. H. Poincar Anal. Non Lint’eaire 1998; 15:459–492. DOI: 10.1016/S0294-

1449(98)80031-0.21. Miranville A, Zelik S. Attractors for dissipative partial differential equations in bounded and unbounded domains. In Handbook of differential equa-

tions: evolutionary equations, Vol. IV. Handb. Differ. Equ.: Elsevier North-Holland, Amsterdam, 2008; 103–200, DOI: 10.1016/S1874-5717(08)00003-0.22. Babin A. Global attractors in PDE. Handbook of dynamical systems, Vol. 1B. Elsevier B.V.: Amsterdam, 2006.23. Bricmont J, Kupiainen A, Taskinen J. Stability of Cahn-Hilliard fronts. Communications on Pure and Applied Mathematics 1999; 52:839–871. DOI:

10.1002/(SICI)1097-0312(199907)52:7.


19

07


24. Korvola T, Kupiainen A, Taskinen J. Anomalous scaling for 3D Cahn-Hilliard Fronts. Communications on Pure and Applied Mathematics 2005;58:1077–1115. DOI: 10.1002/cpa.20055.

25. Caffarelli L, Muler N. An L1 bound for solutions of the Cahn-Hilliard equation. Archive for Rational Mechanics and Analysis 1995; 133(2):129–144.DOI: 10.1007/BF00376814.

26. Abergel F. Attractor for a Navier-Stokes flow in an unbounded domain. Attractors, inertial manifolds and their approximation (Marseille-Luminy,1987). RAIROModel.Math. Anal. Numer. 1989; 23:359–370.

27. Abergel F. Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains. Journal for DifferentialEquations 1990; 83:85–108. DOI: 10.1016/0022-0396(90)90070-6.

28. Babin AV, Vishik MI. Attractors of partial differential equations in an unbounded domain. Proceedings of the Royal Society of Edimburgh 1990;116A:221–243. DOI: 10.1017/S0308210500031498.

29. Efendiev M, Zelik S. The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Communications on Pure and AppliedMathematics 2001; 54:625–688. DOI: 10.1002/cpa.1011.

30. Efendiev M, Miranville A, Zelik S. Global and exponential attractors for nonlinear reaction-diffusion systems in unbounded domains. Proceedings ofthe Royal Society of Edimburgh Section A 2004; 134:271–315. DOI: 10.1017/S030821050000322X.

31. Zelik S. Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains. Journal of Dynamic Differential Equations 2007;19:1–74. DOI: 10.1007/s10884-006-9007-4.

32. Bonfoh A. Finite-dimensional attractor for the viscious Cahn-Hilliard equation in an unbounded domain. Quarterly of Applied Mathematics 2006;64(1):94–104. DOI: 10.1090/S0033-569X-06-00988-3.

33. Zelik S. Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains. In Instability in models connected with fluid flows.II, Vol. 7. Int. Math. Ser. (N,Y.): Springer, New York, 2008; 255–327, DOI: 10.1007/978-0-387-75219-8-6.

34. Zelik S. Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip. Glasg. Math. J. 2007; 49(3):525–588. DOI:10.1017/S0017089507003849.

35. Mielke A, Zelik S. On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms. WIAS, Preprint N 2010:1500.36. Ball J. Global attractors for damped semilinear wave equations, Partial differential equations and applications. Discrete Continuous Dynamical Systems

2004; 10(1-2):31–52. DOI: 10.3934/dcds.2004.10.31.37. Moise I, Rosa R, Wang X. Attractors for non-compact semigroups via energy equations. Nonlinearity 1998; 11(5):1369–1393. DOI: 10.1088/0951-

7715/11/5/012.38. Chepyzhov V, Vishik M, Zelik S. A strong trajectory attractor for a dissipative reaction-diffusion system. Doklady Mathematics 2010; 82(3):869–873.

DOI: 10.1134/S1064562410060086.39. Chepyzhov V, Vishik M, Zelik S. Strong trajectory attractors for dissipative Euler equations. Journal of Mathematics Pures and Applications 2011; (9)

96(4):395–407. DOI: 10.1016/j.matpur.2011.04.007.40. Zelik S. Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity. Communications on Pure and Applied

Mathematics 2003; 56(5):584–637. DOI: 10.1002/cpa.10068.41. Temam R. Infinite-dimensional dynamical systems in mechanics and physics. In Applied Mathematical Sciences, Vol. 68. Springer-Verlag: New York,

1997.42. Babin AV, Vishik MI. Attractors of evolution equations. In Studies in Mathematics and its Applications, Vol. 25. North-Holland Publishing Co.:

Amsterdam, 1992.43. Zelik S. Multiparameter semigroups and attractors of reaction-diffusion equations in Rn . (Russian) Tr. Mosk. Mat. Obs. 2004; 65:114–174, translation

in Trans. Moscow Math. Soc., 2004, 105–160. DOI: 10.1090/S0077-1554-04-00145-1.44. Volpert A, Volpert V, Volpert Vl. Traveling wave solutions of parabolic systems. In Translations of Mathematical Monographs, Vol. 140. American

Mathematical Society: Providence, RI, 1994.45. Ogiwara T, Matano H. Stability analysis in order-preserving systems in the presence of symmetry. Proceedings of the Royal Society of Edimburgh

Section A 1999; 129(2):395–438. DOI: 10.1017/S0308210500021429.46. Eckmann J-P, Rougemont J. Coarsening by Ginzburg-Landau dynamics. Communications in Mathematical PHysics 1998; 199(2):441–470. DOI:

10.1.1.48.5202.47. Zelik S, Mielke A. Multi-pulse evolution and space-time chaos in dissipative systems. Mem. Amer. Math. Soc. 2009; 198(925).48. Chepyzhov VV, Vishik MI. Evolution equations and their trajectory attractors. Journal of Mathematics Pures and Applications 1997; (9) 76(10):913–964.

DOI: 10.1016/S0021-7824(97)89978-3.49. Robertson A, Robertson W. Topological vector spaces. Reprint of the second edition. In Cambridge Tracts in Mathematics, Vol. 53. Cambridge

University Press: Cambridge-New York, 1980.50. Brezis H. Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Vol. 5. North-Holland Mathematical Studies:

North-Holland, Amsterdam, 1973.51. Rocca E, Schimperna G. Universal attractor for some singular phase transition systems. Physics D 2004; 192(3-4):279–307. DOI:

10.1016/j.physd.2004.01.024.1

90

8


Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

Documents

Transcript of Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains