IOP PUBLISHING NONLINEARITY

Nonlinearity 20 (2007) 1773–1797 doi:10.1088/0951-7715/20/8/001

Finite-dimensionality of attractors for degenerateequations of elliptic–parabolic type

A Miranville1 and S Zelik2

1 Universite de Poitiers, Laboratoire de Mathematiques et Applications, UMR 6086, SP2MI,Boulevard Marie et Pierre Curie, 86962 Chasseneuil Futuroscope Cedex, France2 Department of Mathematics Guildford, University of Surrey, GU2 7XH, UK

Received 5 December 2006, in final form 22 May 2007Published 15 June 2007Online at stacks.iop.org/Non/20/1773

Recommended by E S Titi

AbstractOur aim in this paper is to study the long time behaviour, in terms of finite-dimensional attractors, of degenerate triply nonlinear equations. In particular,we are interested in the case where the equation becomes elliptic in some region.

Mathematics Subject Classification: 35B41, 35B45, 35K65

Introduction

In this paper we are interested in the study of the long time behaviour (in terms of finite-dimensional attractors) of triply nonlinear equations of the form

∂tB(u) = div(a(∇xu)) − f (u) + g, (0.1)

in a bounded regular domain of R3. Such equations occur, e.g. in the study of phase separation,

and, in particular, in models of Allen–Cahn equations. More precisely, by considering a localmicroforce balance, Gurtin proposed in [Gu] generalized Allen–Cahn equations of the form

a

(u, ∇xu,

∂u

∂t

)∂u

∂t− κ�xu + f (u) = 0,

where a � 0 and κ is a positive constant. The starting point in these models is the classicalGinzburg–Landau free energy (see [Gu]). Now, if we consider a more general non-isotropicfree energy as in [TC], we obtain, when a only depends on u, a triply nonlinear equation ofthe form (0.1).

The study of equations of the form (0.1) can be found in [AL,Ba,D1,DG,DS,DV,EMR,ER,EfZ2,GM,Mi,R,Ro,S] (actually, some of these works also consider the more general caseof differential inclusions).

Here, we are more particularly interested in the case where the equation is elliptic whenu � 0. Such a situation has been considered, e.g. in [DG,DV,O]. However, there is, to the best

0951-7715/07/081773+25$30.00 © 2007 IOP Publishing Ltd and London Mathematical Society Printed in the UK 1773

1774 A Miranville and S Zelik

of our knowledge, no result concerning the long time behaviour of the solutions of (0.1) in thatcase; more precisely, our aim here is to prove the existence of finite-dimensional attractors.

It is worth noting that, in spite of a very large number of results concerning the finite-dimensionality of attractors (see, e.g. [BV, T] and the references therein), the validity ofany finite-dimensional reduction for equations with singularities or degenerations in theleading terms (such as porous media type equations and elliptic–parabolic problems) hasbeen completely unclear for a long time. The main obstacle here is the lack of regularity (andof smoothing) near the degeneration points, which prevents from using classical methods.Furthermore, as it has recently been established, the problem is far from being just technicaland the degenerations can lead to essentially new types of attractors which are not observablein ‘regular’ equations in bounded domains. Indeed, as shown in [EfZ1], the global attractor ofthe simplest degenerate analogue of the real Ginzburg–Landau equation in a bounded domain�, namely,

∂tu = �x(u3) + u − u3, u

∣∣∂�

= 0,

is infinite-dimensional (to the best of our knowledge, this is the first example of a physicallyrelevant dissipative system in a bounded domain with an infinite-dimensional global attractor).Furthermore, the ‘thickness’ of this attractor (in the sense of Kolmogorov’s ε-entropy) is typicalof Sobolev spaces embeddings and is of the order of that of compact absorbing sets. Thus,this attractor is ‘huge’, even in comparison with the infinite-dimensional global attractors of‘regular’ systems in unbounded domains for which the typical thickness is usually of the orderof spaces of analytic functions embeddings, see, e.g., [Z2] and the references therein.

Nevertheless, a satisfactory finite-dimensional reduction still seems possible under suitablerestrictions on the structure of the equations. Roughly speaking, these conditions shouldprevent the energy income near the degenerations (the equations must be exponentially stablenear all the degeneration points). In particular, this condition is violated for the abovedegenerate Ginzburg–Landau equation, since the ‘linearization’ near u = 0 reads

∂tu = u

and is, obviously, not stable.The validity of the finite-dimensional reduction (in terms of global and exponential

attractors) under such additional restrictions has been verified in [EfZ1] for porous mediaequations. Furthermore, analogous results for degenerate doubly nonlinear equations ofthe form

B(∂tu) = �xu − f (u) + g (0.2)

(here, B degenerates) have recently been obtained in [EfZ2].It is however worth emphasizing that, surprisingly, the quasilinear equation (0.1)

considered in this paper appears to be much more complicated than the, at least formally,more difficult fully nonlinear problem (0.2). Indeed, even classical solutions are available forequation (0.2) with a finite number of ‘reasonable’ degeneration points for B, see [EfZ2]. Incontrast to this, only Holder continuous solutions are to be expected for equation (0.1) witha finite number of degeneration points and, e.g., in the elliptic–parabolic case, discontinuitiesare even to be expected. This lack of regularity prevents from directly applying the techniquesdevised in [EfZ1] and [EfZ2]. In particular, the lack of information on the time derivative ∂tu

in the regions where u � 0 (in the elliptic–parabolic case) is crucial here.Nevertheless, by using some appropriate combination of the results of [MZ] and the

so-called l-trajectories method (see [MP]), we are able to overcome the above-mentioneddifficulties and justify the finite-dimensional reduction under some natural assumptions on

Finite-dimensionality of attractors 1775

B, a and f (point or elliptic–parabolic degenerations for B, standard ellipticity and non-degeneracy assumptions on a, plus some restrictions on f yielding that there is no energyincome near the degenerations, see section 1 for details). Thus, the main result of this paperis the existence (in that case) of finite-dimensional global and exponential attractors for thesemigroup associated with problem (0.1), see theorem 2.2.

This paper is organized as follows. In section 1, we give the main assumptions and provethe existence and uniqueness of solutions. Then, in section 2, we prove the existence of theglobal attractor and, under some additional assumptions on B, we prove, in section 3, theexistence of an exponential attractor, which yields that the global attractor has finite fractaldimension. Finally, we give, in section 4, some remarks and possible extensions.

1. A priori estimates, existence and uniqueness of solutions

We consider the following problem in a bounded smooth domain � ⊂ R3:{

∂tB(u) = div(a(∇xu)) − f (u) + g,

u∣∣∂�

= 0, B(u)∣∣t=0 = b0,

(1.1)

where u = u(t, x) is an unknown function, B, a and f are given functions and g ∈ L∞(�)

corresponds to given external forces.We assume that the nonlinearity a derives from a strictly convex potential A ∈ C2(R3), i.e.{

1. a(z) := ∇zA(z), a(0) = 0,

2. κ1 � A′′(z) � κ2, κ1, κ2 > 0.(1.2)

We also assume that the second nonlinearity f ∈ C1(R) is dissipative,

lim inf|z|→∞

f (z)

z� α0 > 0, (1.3)

and has the following structure:

f (z) = f0(z) + φ(B(z)), (1.4)

where the functions f0 and φ also belong to C1(R) and f0 is monotone,

f ′0(z) � 0. (1.5)

Finally, the third nonlinearity B is assumed to be smooth enough, namely, B ∈ C1(R), withB ′(z) � 0, and to satisfy one of the following classes of assumptions:{

1. B(0) = 0, B(z) �= 0, z �= 0,

2. κ1|z|p � B ′(z) � κ2|z|p, κ1, κ2 > 0; Assumptions (A)

or {1. B(z) = 0, z � 0, B ′(z) > 0, z > 0,

2. κ1|z|p � B ′(z) � κ2|z|p, z � 0, κ1, κ2 > 0; Assumptions (B)

where p � 0 is some fixed number. Thus, when assumptions (A) are satisfied, we havea parabolic system (1.1) with at most one degeneration point for B at z = 0 and, whenassumptions (B) are satisfied, (1.1) is parabolic for u > 0 and elliptic for u � 0.

As usual, in order to study the degenerate case, we approximate problem (1.1) by non-degenerate ones,{

∂tB(u) + ε∂tu = div(a(∇xu)) − f (u) + g,

u∣∣∂�

= 0, (B(u) + εu)∣∣t=0 = b0,

(1.6)

1776 A Miranville and S Zelik

where 0 < ε � 1 is a small parameter. Equation (1.6) is a non-degenerate second-orderparabolic problem which, obviously, has a unique solution u = uε ∈ W(1,2),q([0, T ] × �),for every q < ∞, see, e.g. [LSU] (if B and b0 are smooth enough, say, B, b0 of class C2;the existence of solutions for less regular initial data then follows from the a priori estimatesobtained below and standard approximation arguments). Our aim is now to obtain uniformwith respect to ε a priori estimates on u and then obtain a solution of (1.1) passing to thelimit ε → 0.

We start with a uniform dissipative L∞-estimate for the solutions of (1.6).

Theorem 1.1. Let the above assumptions hold and the initial datum b0 be such that there existsu0 ∈ L∞(�) such that

B(u0) = b0, i.e. b0 ∈ L∞(�), and, when assumptions (B) hold,

we assume that, in addition, b0(x) � 0. (1.7)

Then, the solution u = uε of equation (1.6) satisfies the following estimate:

‖u(t)‖L∞(�) � C(1 + ‖g‖L∞(�)) + Q(‖b0‖L∞(�))e−αt , (1.8)

where the positive constants C and α and the monotonic function Q are independent of ε → 0.

Proof. As usual, the proof of estimate (1.8) is based on the comparison principle. We firstderive the upper L∞-bound on the solution u. To this end, we note that, due to the dissipativityassumption (1.3), there exists a sufficiently large constant K > 0 such that

f (u) � 1/2α0(u − K), u � 0; f (u) � 1/2α0(u + K), u � 0. (1.9)

Let now y = y+(t) be solution of the following first-order ODE:

d

dt(B(y) + εy) + 1/2α0(y − K) = ‖g‖L∞(�), y(0) = max{K, sup

x∈�

u(0, x)}. (1.10)

Then, y(t) � K and, consequently, y(t) is a supersolution for equation (1.6). The comparisonprinciple for the non-degenerate second-order parabolic problem (1.6) reads

u(t, x) � y(t), (t, x) ∈ R+ × �. (1.11)

Using now the fact that, when both (A) and (B)hold, the function B(z) grows monotonicallyas z → +∞, one can easily deduce from (1.10) that

y(t) � C(1 + ‖g‖L∞(�)) + Q(‖b0‖L∞(�))e−αt , t � 0, (1.12)

for suitable positive constants C and α and monotonic function Q which are independent ofε. This gives the upper L∞-bound on the solution u of the form (1.8).

We now check the lower bound. Arguing analogously, we establish that the solutiony = y−(t) of the following ODE:

d

dt(B(y) + εy) + 1/2α0(y + K) = −‖g‖L∞(�), y(0) = min{−K, inf

x∈�u(0, x)}, (1.13)

gives a subsolution of problem (1.8) if K is large enough and we have

u(t, x) � y(t), (t, x) ∈ R+ × �. (1.14)

Then, when assumptions (A) hold, the situation is completely analogous to the previous caseand we have the analogue of (1.12) for the solution −y(t), which gives (1.8) and finishes theproof in that case.

Finite-dimensionality of attractors 1777

Let us now assume that conditions (B) hold. In that case, we have B(y(t)) ≡ 0, sincey(t) � −K < 0. Moreover, due to (1.7), we have b0 � 0, which, in turn, implies that u(0) � 0and y(0) = −K . So, (1.13) reads

εd

dty(t) + 1/2α0(y(t) + K) = −‖g‖L∞(�), y(0) = −K, (1.15)

which can be solved explicitly,

y(t) = −K − ‖g‖L∞(�)

(1 − e− α0

2εt)

, t � 0.

Thus, y(t) � −K − ‖g‖L∞(�), ∀t � 0, and estimate (1.8) is also verified underassumptions (B). This finishes the proof of theorem 1.1.

Our next aim is to obtain uniform estimates on the derivatives of u. We state them in threesimple lemmas below.

Lemma 1.1. Let the assumptions of theorem 1.1 hold. Then, the solution u of (1.6) satisfies∫ t+1

t

‖∇xu(s)‖2L2(�) ds � C(1 + ‖g‖L∞(�)) + Q(‖b0‖L∞(�))e

−αt , (1.16)

where the positive constants C and α and the monotonic function Q are independent of ε and t .

Proof. Multiplying equation (1.6) by u and integrating over [t, t + 1] × �, we have

(B(u(t + 1)) − B(u(t)), 1)L2(�) + ε/2[‖u(t + 1)‖2L2(�) − ‖u(t)‖2

L2(�)]

+∫ t+1

t

(a(∇xu(s)), ∇xu(s))L2(�) ds

=∫ t+1

t

[(g, u(s))L2(�) − (f (u(s)), u(s))L2(�)] ds, (1.17)

where B(v) := ∫ v

0 B ′(u)u du. Using now the L∞-estimates for u obtained in the previoustheorem and the fact that a = ∇uA, for a strictly convex potential A (see assumptions 1.2), weobtain (1.16) and finish the proof of the lemma.

The next lemma gives a gradient-like energy inequality.

Lemma 1.2. Let the above assumptions hold. Then, the solution u of problem (1.6) satisfiesthe following estimates:∫ t+1

t

[(B ′(u(s))∂tu(s), ∂tu(s))L2(�) + ε/2‖∂tu(s)‖2L2(�)] ds + ‖∇xu(t)‖2

L2(�)

� t + 1

t

(C(1 + ‖g‖2

L∞(�)) + Q(‖b0‖L∞(�))e−αt

), t > 0, α > 0, (1.18)

where all the constants and the monotonic function Q are independent of ε. If, in addition,u(0) ∈ W

1,20 (�), then

∫ 1

0[(B ′(u(s))∂tu(s), ∂tu(s))L2(�) + ε‖∂tu(s)‖2

L2(�)] ds + ‖∇xu(t)‖2L2(�)

� C(

1 + ‖g‖2L∞(�) + Q(‖b0‖L∞(�)) + ‖∇xu(0)‖2

L2(�)

), 0 � t � 1. (1.19)

1778 A Miranville and S Zelik

Proof. Multiplying equation (1.6) by (t − T )∂tu and integrating over [T , T + 2] ×�, we have∫ T +2

T

[(s − T )(B ′(u(s))∂tu(s), ∂tu(s))L2(�) + ε/2‖∂tu(s)‖2L2(�)] ds

+ (t − T )‖∇xu(t)‖2L2(�) + (t − T )(F (u(t)), 1)L2(�)

� C

∫ T +2

T

[‖∇xu(s)‖2L2(�) + (F (u(s)), 1)L2(�)] ds + C‖g‖2

L∞(�), (1.20)

for t ∈ [T , T +2]. This internal estimate, together with the dissipative estimates for ‖u(t)‖L∞(�)

and ‖u‖L2([T ,T +1],W 1,2(�)) obtained in theorem 1.1 and lemma 1.1, respectively, give the desiredestimate (1.18). Estimate (1.19) can be proven analogously, but is much simpler, since weonly need to multiply the equation by ∂tu. This finishes the proof of lemma 1.2.

In the third lemma, we state the W 2,2-regularity result.

Lemma 1.3. Let the above assumptions hold. Then, the solution u(t) of problem (1.6) satisfiesthe following estimate:∫ t+1

t

‖u(s)‖2W 2,2(�) ds � t + 1

t

(C(1 + ‖g‖2

L∞(�)) + Q(‖b0‖L∞(�))e−αt

), (1.21)

t > 0, where C, α > 0 and Q are independent of ε.

Proof. We rewrite (1.6) as an elliptic boundary value problem for every fixed t ,

div(a(∇xu(t))) = ∂tB(u(t)) + ε∂tu(t) + f (u(t)) − g := Hu(t), u(t)∣∣∂�

= 0. (1.22)

Then, according to the H 2–L2-regularity result for second-order quasilinear elliptic equations(see, e.g., [Mi]), we have

‖u(t)‖W 2,2(�) � C‖Hu(t)‖L2(�), (1.23)

where the constant C is independent of u. Using now estimates (1.8) and (1.18) to estimatethe L2-norm of Hu, we obtain (1.21) and finish the proof of the lemma.

Finally, we formulate an L1-Lipschitz continuity result based on the Kato inequality (see[K] for the original inequality (�xu, sgnu) � 0 and, e.g. [Ko] and [Si] for more general second-order elliptic operators, in particular for inequalities of the form (div(α(x)∇xu), sgn u) � 0).

Lemma 1.4. Let the above assumptions hold and let u1(t) and u2(t) be two solutions ofproblem (1.6). Then, the following estimates hold:

‖B(u1(t)) − B(u2(t))‖L1(�) + ε‖u1(t) − u2(t)‖L1(�)

� CeKt(‖B(u1(0)) − B(u2(0))‖L1(�) + ε‖u1(0) − u2(0)‖L1(�)

), (1.24)

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

Proof. We set v(t) = u1(t) − u2(t). This function solves

∂t [B(u1(t)) − B(u2(t)) + εv(t)] = div[a(∇xu1(t)) − a(∇xu2(t))]

−[f0(u1(t)) − f0(u2(t))] − [φ(B(u1(t)) − φ(B(u2(t))]. (1.25)

Multiplying this equation by sgnv(t) = sgn(B(u1(t))−B(u2(t))) and using the Kato inequality(this multiplication can be easily justified in a standard way, since (1.6) is non-degenerate andthe solutions u1 and u2 are sufficiently regular), we have

∂t [‖B(u1(t)) − B(u2(t))‖L1(�) + ε‖v(t)‖L1(�)]

+ (f0(u1(t)) − f0(u2(t)), sgn(u1(t)) − sgn(u2(t)))L2(�)

� (φ(B(u1(t))) − φ(B(u2(t))), sgn(B(u1(t)) − B(u2(t))))L2(�). (1.26)

Finite-dimensionality of attractors 1779

Using now assumptions (1.4) and (1.5) for the functions f0 and φ, together with the L∞-boundsfor u1 and u2, we deduce that

∂t [‖B(u1(t)) − B(u2(t))‖L1(�) + ε‖v(t)‖L1(�)] � K‖B(u1(t)) − B(u2(t))‖L1(�),

which, together with the Gronwall inequality, give (1.24) and finish the proof of the lemma.

We are now able to formulate the solvability result for the limit degenerate problem (1.1),which can be considered as the main result of this section.

Theorem 1.2. Let the assumptions of theorem 1.1 hold. Then, for every b0 ∈ L∞(�), b0 � 0,problem (1.1) has at least one solution u(t) belonging to the following class:

u ∈ L∞([0, T ] × �), B(u) ∈ C([0, T ], L1(�)),

u ∈ L∞([0, T ], W 1,20 (�)) ∩ L2([t, T ], W 2,2(�)),

∂tR(u) ∈ L2([t, T ] × �), t > 0, R(v) :=∫ v

0

√B ′(u) du. (1.27)

Furthermore, this solution satisfies all the estimates obtained in lemmas 1.1–1.3 and theorem1.1 and can be obtained in a unique way as the limit of the corresponding solutions uε of theregularized problems (1.6) as ε → 0. Finally, for every two such solutions u1(t) and u2(t)

(corresponding to different initial data b10 and b2

0), the following global L1-Lipschitz continuityholds:

‖B(u1(t)) − B(u2(t))‖L1(�)

� CeKt‖B(u1(0)) − B(u2(0))‖L1(�) = CeKt‖b10 − b2

0‖L1(�), (1.28)

where the constants C and K only depend on the L∞-norms of b10 and b2

0.

Proof. Let un(t) := uεn(t) be a sequence of solutions of the approximate problems (1.6) with

εn → 0 and with the same initial datum b0. Then, due to theorem 1.1 and lemmas 1.1–1.3, wecan assume, without loss of generality, that

un → u weakly-∗ in

L∞([0, T ] × �) ∩ L∞([t, T ], W 1,20 (�)) ∩ L2([t, T ], W 2,2(�)), t > 0. (1.29)

The main problem is, however, that, when assumptions (B) hold, we do not control the timederivative ∂tu in the region u � 0 and, consequently, we cannot directly extract the strongconvergence un → u in a suitable space from (1.29) (which is essential for the passage to thelimit n → ∞ in the nonlinear terms of equation (1.6)). In order to overcome this difficulty, weuse monotonicity arguments. We first note that lemma 1.2 allows one to control the L2-norm ofthe time derivative of the functions ψn(t) := B(un(t)) on every interval [t, T ]. Furthermore,its x-gradient can also be easily controlled, since ‖∇xun(t)‖L2(�) is uniformly bounded on[t, T ]. Thus, the sequence ψn is precompact in the strong topology of L2([t, T ] × �) and,without loss of generality, we can assume, in addition, that

ψn → ψ strongly in C([t, T ], L2(�)). (1.30)

Let us prove that

ψ = B(u). (1.31)

To this end, we use the standard fact that the operator z �→ B(z) is maximal monotone inL2([t, T ]×�), since B ′(z) � 0 (being pedants, we should first cut off the function B for largez in order to make it well-defined as an operator in L2([t, T ] × �), but, since we control the

1780 A Miranville and S Zelik

L∞-norm of the solutions, this procedure is not essential and is omitted). Thus, in order toverify (1.31), we only need to check that

(ψ − B(w), u − w)L2([t,T ]×�) � 0, ∀w ∈ L2([t, T ] × �), (1.32)

see, e.g. [Li]. It remains to be noted that the strong convergence (1.30) allows (1.32) to beobtained by a direct passage to the limit n → ∞ in the following obvious inequality:

(B(un) − B(w), un − w)L2([t,T ]×�) = (ψn − B(w), un − w)L2([t,T ]×�) � 0. (1.33)

Thus, (1.31) is verified and, consequently,

B(un) → B(u) strongly in C([t, T ], L2(�)),

which, in turn, implies that

∂tB(un) → ∂tB(u) weakly inL2([t, T ] × �),

φ(B(un)) → φ(B(u)) strongly in C([t, T ], L2(�)). (1.34)

Moreover, arguing analogously, we have

R(un) → R(u) strongly in C([t, T ], L2(�)). (1.35)

Consequently, ∂tR(un) = √B ′(un)∂tun → ∂tR(u) weakly in L2([t, T ] × �) and

‖∂tR(u)‖L2([t,T ]×�) � lim infn→∞ ‖∂tR(un)‖L2([t,T ]×�). (1.36)

In order to pass to the limit in the right-hand side of (1.6), we need the following lemma.

Lemma 1.5. Let the above assumptions hold and let un and u be as above. Then,

∂tB(u) = ∂tR(u) ·√

B ′(u) (1.37)

and, for every t > 0,

limn→∞(∂tB(un), un)L2([t,T ]×�) = (∂tB(u), u)L2([t,T ]×�). (1.38)

Proof of the lemma. Since ∂tun is regular enough, we have

∂tB(un) = ∂tR(un) ·√

B ′(un). (1.39)

We now recall that the weak convergences ∂tB(un) → ∂tB(u) and ∂tR(un) → ∂tR(u)

have already been established. Thus, (1.37) will be proven provided that we check that√B ′(un) → √

B ′(u) strongly in L2([t, T ]×�). Let us first assume that assumptions (A) hold.Then, the inverse function v �→ B−1(v) exists and is even Holder continuous. Consequently,the strong convergence of B(un) to B(u) implies the strong convergence of un to u and,therefore,

√B ′(un) also converges strongly to

√B ′(u), which, in turn, implies (1.37). Let now

assumptions (B) be satisfied. Then, since B(u) ≡ 0 for u � 0 and is strictly monotonefor u > 0, we have a Holder continuous partial inverse function v �→ T (v) such thatT (B(u)) = u+ := max{u, 0}. Thus, in that case, the strong convergence B(un) to B(u)

only implies that u+n converges strongly to u+. Nevertheless, since now B ′(u) = B ′(u+), this

convergence is sufficient to conclude that√

B ′(un) converges strongly to√

B ′(u) and finishthe proof of equality (1.37) for both assumptions (A) and (B).

In order to check (1.38), it is now sufficient to rewrite it in the form

limn→∞(∂tR(un),

√B ′(un) · un)L2([t,T ]×�) = (∂tR(u),

√B ′(u) · u)L2([t,T ]×�)

and note that, analogously to the arguments given above,√

B ′(un) · un converges strongly to√B ′(u) · u. This finishes the proof of lemma 1.5.

Finite-dimensionality of attractors 1781

It is now not difficult to finish the passage to the limit n → ∞ in equations (1.6) for un

and verify that u solves indeed the limit degenerate problem (1.1). To this end, we use thestandard fact that the quasilinear differential operator

A(u) := −div(a(∇xu)) + f0(u) (1.40)

is maximal monotone in L2([t, T ], W 1,20 (�)) (we recall that f0 is monotone). Then, we rewrite

equation (1.6) in the form

A(un) = θn := g − ∂tB(un) − φ(B(un)). (1.41)

According to the above convergences, we have

∇xun → ∇xu, θn → θ := g − ∂tB(u) − φ(B(u)) weakly in L2([t, T ] × �).

Moreover, using (1.34) and (1.38), we see that

limn→∞(θn, un)L2([t,T ]×�) = (θ, u)L2([t,T ]×�),

which, by monotonicity arguments, implies that A(u) = θ . Thus, the function u solves indeedthe limit degenerate problem (1.1).

Passing to the limit n → ∞ in the estimates of theorem 1.1 and lemmas 1.1–1.4, it followsthat these estimates hold for the solution of the limit problem as well.

Thus, there only remains to check the uniqueness and the fact that the limit solution u issuch that B(u) ∈ C([0, T ], L1(�)). To this end, we take the difference between equations (1.6)for un and um, respectively, multiply the resulting equation by sgn(un − um), use the fact thatun(0) = um(0) and argue as in lemma 1.4 to infer

∂t‖B(un(t)) − B(um(t))‖L1(�) � K‖B(un(t)) − B(um(t))‖L1(�)

+C(εn + εm)(‖∂tun(t)‖L1(�) + ‖∂tum(t)‖L1(�)). (1.42)

Assume first that b0 is chosen in such a way that, in addition, u(0) ∈ W1,20 (�). Then, according

to estimates (1.18) and (1.19), we can control the derivatives in the right-hand side of (1.42)and, using the Gronwall inequality, deduce that

‖B(un(t)) − B(um(t))‖L1(�) � CeKt (εn + εm)1/2. (1.43)

Thus, B(un) is a Cauchy sequence in C([0, T ], L1(�)) and, consequently, B(u) =limn→∞ B(un) belongs to C([0, T ], L1(�)) and is determined in a unique way by the solutionsof the approximate equations (1.6). In the general case, i.e. b0 ∈ L∞(�), b0 � 0, it is sufficientto approximate u(0) by smooth initial data un(0) in L1(�) and pass to the limit n → ∞. Thisyields that B(u) ∈ C([0, T ], L1(�)) for general initial data as well and finishes the proof ofthe theorem.

Remark 1.1.

(i) It is worth emphasizing that we have proven the uniqueness of a solution u only in thesubclass of (1.27) of the solutions which can be obtained by passing to the limit ε → 0in the non-degenerate approximate equations. The uniqueness of a solution in the wholeclass (1.27) is much more delicate, since, for degenerate equations, the validity of theKato inequality is non-trivial and must be verified. Since, everywhere in the sequel, wewill only consider the solutions of equation (1.1) which can be obtained by the above limitprocedure, this uniqueness is not important for what follows and we refer the reader to[O] for a more detailed exposition.

1782 A Miranville and S Zelik

(ii) We also mention that we only consider initial data b0 ∈ L∞(�) in order to excludefrom the very beginning the ‘pathological’ singular solutions which may appear in doublynonlinear equations with less regular initial data, see, e.g. [D2] and [EfZ2]. Moreover, itis worth noting that our assumption B ′(z) ∼ zp is necessary only near the degenerationpoint z = 0 and should not be considered as some growth assumption as z → ∞, forwhich we only need B ′(z) � C > 0.

We conclude this section by some kind of additional regularity for the time derivative ∂tu

which will be crucial for our theory.

Proposition 1.1. Let the above assumptions hold and let u(t) be a solution of (1.1) asconstructed in the previous theorem. Assume also that the nonlinearity B belongs to C2(R)

and satisfies the additional condition

|B ′′(z)|4/3 � CB ′(z), z ∈ I, ∀I ⊂ R bounded. (1.44)

Then, there exists a positive constant τ , 0 < τ < 1, depending only on the L∞-norm of theinitial datum b0, such that, for every time interval [T , T + 1], there exists T0 ∈ [T , T + 1](depending on the solution u) such that

‖∂tR(u(t))‖2L2(�) +

∫ T0+τ

T0

‖∂t∇xu(s)‖2L2(�) ds � Q(‖b0‖L∞(�)), (1.45)

for all t ∈ [T0, T0 + τ ], where the monotonic function Q is also independent of the concretechoice of u.

Proof. Below we only give the formal derivation of estimate (1.44), which can be justified in astandard way by considering the approximate solutions of (1.6) and passing to the limit ε → 0.

We first note that, according to lemma 1.2, we have∫ T +1

T

(B ′(u(s))∂tu(s), ∂tu(s))L2(�) ds � Q(‖b0‖L∞(�)), T � 1/2.

Consequently, (B ′(u(t))∂tu(t), ∂tu(t))L2(�) is finite for almost all t and, for every time interval[T , T + 1], there exists at least one point T0 = T0(u, T ) ∈ [T , T + 1] such that

(B ′(u(T0))∂tu(T0), ∂tu(T0))L2(�) � 2Q(‖b0‖L∞(�)). (1.46)

Since equation (1.1) is autonomous, then, without loss of generality, we may assume thatT0 = 0.

We now differentiate equation (1.1) with respect to t and set v = ∂tu. Then, we have

B ′(u(t))∂tv(t) + B ′′(u(t))|v(t)|2 = div(a′(∇xu(t))∇xv(t))

−f ′0(u(t))v(t) − φ′(B(u(t))B ′(u(t))v(t).

Multiplying this equation by v, integrating with respect to x ∈ � and using the fact that a andf are monotonic, we have

∂t (B′(u(t))v(t), v(t))L2(�) + 2θ‖∇xv(t)‖2

L2(�)

� C(B ′(u(t))v(t), v(t))L2(�) + (|B ′′(u(t))|, |v(t)|3)L2(�), (1.47)

for some positive constant θ . Let Iu(t) := (B ′(u(t))v(t), v(t))L2(�). Then, using theorem 1.1,assumption 1.44 and the Sobolev embedding W 1,2(�) ⊂ L6(�), we can estimate the last termin the right-hand side of (1.47) as follows:

(|B ′′(u)|, |v|3)L2(�) = (|B ′′(u)| · |v|3/2, |v|3/2)L2(�)

� (|B ′′(u)|4/3, |v|2)3/4L2(�)

‖v‖3/2L6(�)

� C(B ′(u), |v|2)3/4L2(�)

‖∇xv‖3/2L2(�)

� θ‖∇xv‖2L2(�) + CI 3

u ,

Finite-dimensionality of attractors 1783

where C = C(‖b0‖L∞(�)). Thus, (1.47) reads

∂t Iu(t) + θ‖∇xv(t)‖2L2(�) � C(Iu(t) + Iu(t)

3). (1.48)

Moreover, due to (1.46) and owing to the fact that T0 = 0, we have

0 � Iu(0) � 2Q(‖b0‖L∞(�)). (1.49)

We can note that the differential inequality (1.48) is not strong enough in order to obtain globalin time estimates for Iu(t). Nevertheless, it is sufficient for the required local in time ones.Indeed, due to the comparison principle, we have

Iu(t) � y(t), (1.50)

where y solves

y ′ = C(y + y3), y(0) = 2Q(‖b0‖L∞(�)). (1.51)

Therefore, the local solvability result for the ODE (1.51) gives the existence of a time interval[0, τ ], with τ > 0 only depending on ‖b0‖L∞(�), such that

Iu(t) � y(t) � Q1(‖b0‖L∞(�)), (1.52)

where Q1 is also independent of the concrete choice of u. Integrating now (1.48) with respectto t ∈ [0, τ ] and using (1.52), we deduce the required estimate for the integral norm of ∇xv

and finish the proof of proposition 1.1.

Remark 1.2. Obviously, assumption 1.44 is automatically satisfied in the non-degenerate case(which corresponds to assumptions (A) and p = 0). However, in the degenerate case p > 0,this gives rather essential restrictions on the regularity of the function B near the degenerationpoint. In particular, it is not difficult to verify that (1.44) implies that p � 4 if p > 0.This assumption will be satisfied, e.g. if the function B is of class C5 near the degenerationpoint z = 0.

2. Semigroups and attractors

In this section, we show that the semigroup associated with the degenerate equation (1.1)possesses the global attractor in an appropriate phase space and formulate the main result ofthe paper, namely, the existence of an exponential attractor for this semigroup, which will beproven in the next section.

We first define the phase space � for problem (1.1) as follows:

� := {b0 ∈ L∞(�), and, when assumptions (B) hold, b0(x) � 0, x ∈ �, also} (2.1)

and we define the semigroup S(t) associated with equation (1.1) by the following naturalexpression:

S(t)b0 := B(u(t)), where u(t) solves (1.1) with B(u(0)) = b0. (2.2)

Remark 2.1. We see that, in contrast to the usual situation, the semigroup S(t) does not mapu(0) onto u(t), but B(u(0)) onto B(u(t)). This naturally reflects the fact that the solution u(t) isuniquely defined by B(u(0)) and that the equation may become elliptic in some regions; whenassumptions (A) hold, we can actually consider the usual framework. We also emphasizeonce more that, by a ‘solution’ of equation (1.1), we always mean a solution constructedin theorem 1.2 by the limit procedure, no matter whether or not problem (1.1) has other‘pathological’ solutions which are automatically dropped out of our analysis.

We now recall the definition of the global attractor for the semigroup S(t) adapted to ourframework.

1784 A Miranville and S Zelik

Definition 2.1. A set A ⊂ � is the global attractor for the semigroup S(t) associated with thedegenerate problem (1.1) if

(1) it is compact in L1(�) and bounded in L∞(�),(2) it is strictly invariant, i.e. S(t)A = A, ∀t � 0,(3) it attracts the images of all bounded (in the L∞-topology) subsets of � in the topology of

L1(�), i.e. for every bounded subset B of � and every neighbourhood O(A) of the set Ain L1(�), there exists T = T (B, A) such that

S(t)B ⊂ O(A), for all t � T . (2.3)

Remark 2.2.

(i) According to definition 2.1, the attractor A attracts the bounded subsets of � = L∞(�) inthe weaker topology of L1(�) and, thus, coincides with the so-called (L1(�), L∞(�))-attractor in the terminology of Babin and Vishik, see [BV]. We also note that, sincethe trajectories of S(t) are bounded in L∞(�), the space L1(�) in the formulation ofthe attraction property can be replaced by Lp(�), for every finite p. However, the casep = ∞, which coincides with the standard definition of the global attractor, is moredelicate and requires estimates on the solutions of the degenerate system (1.1) in Holderspaces which, to the best of our knowledge, are not known for the elliptic–parabolicproblem when assumptions (B) hold.

(ii) The attraction property can also be formulated via the Hausdorff semi-distance betweensubsets of �. More precisely, let

distV (X, Y ) := supx∈X

infy∈Y

‖x − y‖V (2.4)

be the non-symmetric Hausdorff distance between X and Y in the Banach space V . Then,the attraction property reads: for every bounded subset B ⊂ �,

limt→∞ distL1(�)(S(t)B, A) = 0. (2.5)

The next theorem gives the existence of a global attractor for the semigroup S(t) associatedwith the degenerate problem (1.1).

Theorem 2.1. Let the assumptions of theorem 1.2 hold. Then, the semigroup S(t) definedby (2.2) possesses the global attractor A in the sense of definition 2.1 which is bounded inL∞(�) ∩ W 1,2(�) and possesses the following standard description:

A = B(K∣∣t=0), (2.6)

where K ⊂ L∞(R × �) is the set of all solutions of (1.1) which are defined for all t ∈ R andare globally bounded.

Proof. According to standard results on the existence of the global attractor (see, e.g. [BV]and [T]), we need to check that

(1) the semigroup S(t) is continuous in the L1-topology on every bounded subset of �;(2) the semigroup S(t) possesses an absorbing set which is bounded in L∞(�) and compact

in L1(�).

Finite-dimensionality of attractors 1785

The first assumption is an immediate corollary of the global Lipschitz continuity of thesemigroup S(t), see estimate (1.28). Moreover, it follows from estimates (1.8) and (1.18) thatthe R-ball in the space L∞(�) ∩ W 1,2(�) is an absorbing set for the semigroup S(t) if R islarge enough. Since this ball is, obviously, compact in the topology of L1(�), the existenceof the global attractor A follows, see [BV] and [T]. Its boundedness is now a consequence ofthe fact that the global attractor is contained in any absorbing set and description (2.6) followsfrom the standard description of the global attractor via bounded complete trajectories of theassociated semigroup, see [BV]. This finishes the proof of theorem 2.1.

Our next task is to establish the existence of an exponential attractor for the semigroupS(t) associated with equation (1.1), which implies, in particular, the finite-dimensionality ofthe global attractor constructed in the previous theorem. We first give the definition of anexponential attractor adapted to our framework.

Definition 2.2. A set M ⊂ � is an exponential attractor for the semigroup S(t) associatedwith problem (1.1) if the following conditions are satisfied:

(1) it is bounded in � and compact in L1(�);(2) it is semi-invariant, S(t)M ⊂ M, ∀t � 0;(3) it has finite fractal dimension in L1(�),

dimf (M, L1(�)) � C < ∞;(4) it attracts exponentially the images of all bounded subsets of �, i.e. there exists a positive

constant α and a monotonic function Q such that, for every bounded subset B of the phasespace �, there holds

distL1(�)(S(t)B, M) � Q(‖B‖L∞(�))e−αt , (2.7)

for all t � 0.

The following theorem can be considered as the main result of this paper.

Theorem 2.2. Let the assumptions of theorem 1.1 hold and let, in addition, the nonlinearity B

belong to C2(R) and satisfy assumption (1.44). Then, the semigroup S(t) associated with thedegenerate problem (1.1) possesses a finite-dimensional exponential attractor M in the senseof Definition 2.2 which is bounded in L∞(�) ∩ W 1,2(�).

The proof of this theorem will be completed in the next section. In the remaining of thissection, we formulate an abstract result on the existence of an exponential attractor which isclose to that given in [MZ] (see also [EfMZ]) and is the main technical tool to prove theorem 2.2.

Proposition 2.1. Let H1 and H, H1 ⊂ H, be two Banach spaces such that the embeddingH1 ⊂ H is compact and let C be a closed bounded subset of H. Assume also that there existsa map S : C → C which satisfies the following properties:

(a) it is globally Lipschitz continuous on C, i.e. for every c1, c2 ∈ C, there holds

‖Sc1 − Sc2‖H � L‖c1 − c2‖H, (2.8)

where the Lipschitz constant L is independent of the choice of c1 and c2 belonging to C;(b) there exists an integerN0 such that, for every c ∈ C, there existsn = n(c) ∈ {0, · · · , N0−1}

such that, for every c1 ∈ C, there holds

‖Sc1 − Sc2‖H1 � K‖c1 − c2‖H, c2 := S(n)c, (2.9)

where the discrete semigroup generated on C by the iterations of S is denoted by {S(l),l ∈ N} and the constant K is independent of c and c1.

1786 A Miranville and S Zelik

Then, the discrete semigroup S(l) possesses an exponential attractor M on C, i.e. there existsa set M ⊂ C which satisfies the following properties:

(1) it is a compact subset of C;(2) it is semi-invariant, S(l)M ⊂ M, ∀l ∈ N;(3) it has finite fractal dimension in H,

dimf (M, H) � C1; (2.10)

(4) it attracts exponentially the images of C in the metric of H,

distH(S(l)C, M) � C2e−αl, ∀l ∈ N. (2.11)

Moreover, the positive constants C1, C2 and α can be expressed explicitly in terms of K , L,N0 and some qualitative characteristics of the embedding H1 ⊂ H.

The proof of this proposition repeats word by word that given in [MZ] and is thereforeomitted.

3. Proof of the main result

In this section, we complete the proof of theorem 2.2 and establish the existence of a finite-dimensional exponential attractor for the semigroup S(t) associated with the degenerateequation (1.1). To this end, we need the following result.

Proposition 3.1. Let the assumptions of theorem 2.2 hold, let u be a solution of problem (1.1)and let [T0, T0 + µ] belong to one of the regularity intervals found in proposition 1.1. Thelatter means that, on this time interval, we can control the L2-norm of ∂t∇xu by (1.44). Then,for every other solution u(t), t � T0, of equation (1.1), the following estimate holds:∫ T0+µ

T0+µ/2‖u(t) − u(t)‖2

W 1,2(�) dt � C‖B(u(T0)) − B(u(T0))‖2L1(�), (3.1)

where the constant C only depends on µ and the L∞-norms of u and u and is independent ofthe concrete choice of u and u.

Proof. As in proposition 1.1, we only give below the formal derivation of estimate (3.1), whichcan be easily justified by using the approximate equations (1.6).

We set v(t) := u(t) − u(t). Then, this function obviously solves

∂t [B(u(t)) − B(u(t))] = div[a(∇xu(t)) − a(∇xu(t))] − [f (u(t)) − f (u(t))]. (3.2)

Multiplying this equation by v(t), integrating with respect to x ∈ � and using the monotonicityof a and assumption (1.4), we have

(∂t (B(u(t)) − B(u(t))), v(t))L2(�) + 1/2(a(∇xu) − a(∇xu), ∇xv)L2(�)

+ 2θ‖∇xv(t)‖2L2(�) � C(|B(u(t)) − B(u(t))|, |v(t)|)L2(�), θ > 0. (3.3)

The right-hand side of (3.3) can be estimated as follows:

(|B(u(t)) − B(u(t))|, |v(t)|)L2(�)

= (|B(u(t)) − B(u(t))|1/2, |B(u(t)) − B(u(t))|1/2 · |v(t)|)L2(�)

� C‖B(u(t)) − B(u(t))‖1/2L1(�)

‖v(t)‖3/2L3(�)

� θ‖∇xv(t)‖2L2(�) + C1‖B(u(t)) − B(u(t))‖2

L1(�). (3.4)

Finite-dimensionality of attractors 1787

In order to transform the left-hand side of (3.3), we use the following identity:

∂t [B(u(t)) − B(u(t))] · v(t) = ∂tIu,u(t) + ∂tu(t) · Ju,u(t), (3.5)

where

Iu,u(t) := G(u(t)) − G(u(t)) + B(u(t))v(t),

Ju,u(t) := B(u(t)) − B(u(t)) − B ′(u(t))v(t),(3.6)

with G(v) := ∫ v

0 B(u) du. Inequality (3.3) reads, in view of (3.4) and (3.5),

∂t (Iu,u(t), 1)L2(�) + θ‖∇xv(t)‖2L2(�)

� C‖B(u(t)) − B(u(t))‖2L1(�) + C(|∂tu(t)|, |Ju,u(t)|)L2(�). (3.7)

In order to estimate the terms I and J , we need the following lemma.

Lemma 3.1. Let the above assumptions hold. Then, the functions Iu,u and Ju,u satisfy thefollowing estimates:

1. Iu,u � 0,

2. |Ju,u| � CI1/2u,u · |u − u|,

3. Iu,u � C|B(u) − B(u)|1/2 · |u − u|3/2,

(3.8)

where |u| + |u| � R and the constant C = CR depends on R, but is independent of u, u ∈ R.

Proof. Since G is of class C2 and G′′(z) = B ′(z) � 0, we have

Iu,u =∫ 1

0[B(su + (1 − s)u) − B(u)] ds · (u − u)

=∫ 1

0

∫ 1

0B ′(s1(su + (1 − s)u) + (1 − s1)u) ds1 ds · |u − u|2 � 0. (3.9)

Thus, (3.8)1 is verified. Let us now check (3.8)2 and (3.8)3. Let first assumptions (A) forthe nonlinearity B be satisfied. Then, since B ′(u) ∼ |u|p, estimate (3.9) can be rewritten asfollows:

C2(B′(u) + B ′(u)) · |u − u|2 � Iu,u

� C(|u|p + |u|p)| · |u − u|2 � C1(B′(u) + B ′(u)) · |u − u|2, (3.10)

see [Z1] for details. Analogously, using, in addition, (1.44), we can estimate Ju,u as follows:

|Ju,u| =∫ 1

0

∫ 1

0|B ′′(s1(su + (1 − s)u) + (1 − s1)u)| ds1 ds · |u − u|2

� C

∫ 1

0

∫ 1

0|B ′(s1(su + (1 − s)u) + (1 − s1)u)|3/4 ds1 ds

� C1(B′(u) + B ′(u))3/4 · |u − u|2 (3.11)

and, concerning the difference B(u) − B(u), we have

|B(u) − B(u)| � C(B ′(u) + B ′(u)) · |u − u|. (3.12)

Since 1/2 < 3/4, estimates (3.10)–(3.12), together with the fact that |u| + |u| � R, implyestimates (3.8)2 and (3.8)3. Thus, when assumptions (A) hold, lemma 3.1 is proven.

Let us now consider assumptions (B). To this end, we note that, if u > 0 and u > 0, wehave exactly the same situation as with assumptions (A), so that all the estimates of the lemmaare already verified. The case u < 0 and u < 0 is also obvious since, in that case, both sides

1788 A Miranville and S Zelik

of inequalities (3.8) are identically equal to zero. So, we only need to consider the followingtwo cases:

(1) u > 0 and u < 0;(2) u < 0 and u > 0.

Let us consider case (1). Then, (3.8) reads{2. B ′(u)(u − u) − B(u) � G(u)1/2 · |u − u|,3. G(u) � B(u)1/2 · |u − u|3/2.

(3.13)

We note that, in that case, |u − u| � |u|. Moreover, since B ′(u) ∼ |u|p, p � 0, then,G(u) � CB(u)1/2u3/2 near u = 0, which implies (3.13)3. In order to verify (3.13)2, it sufficesto note that (1.44) implies

B ′(u) � C[B ′(u)]3/4 · |u|, G(u) � CB ′(u) · |u|2. (3.14)

This, together with the fact that |u − u| � |u|, imply (3.13)2. Thus, lemma 3.1 is also verifiedin case (1).

Let us now consider case (2). In that case, (3.8) reads{2. B(u) � C[B(u)(u − u) − G(u)]1/2 · |u − u|,3. B(u)(u − u) − G(u) � B(u)1/2 · |u − u|3/2.

(3.15)

Since B ′(u) ∼ up and u − u � u, we have

B(u)(u − u) − G(u) � B(u)u − G(u) =∫ 1

0sB ′(su) ds · u2 � Cup+2 � C1G(u). (3.16)

Moreover, analogously to (3.14),

C1[B ′(u)]3/4u2 � B(u) � C2[B ′(u)]3/4u2,

C3B′(u)u � B(u) � C4B

′(u)u, C5B′(u)u2 � G(u) � C6B

′(u)u2. (3.17)

Estimates (3.16) and (3.17) imply (3.15). Thus, estimates (3.8) are verified in all cases andlemma 3.1 is proven.

It is now not difficult to finish the proof of the proposition. To this end, we multiplyequation (3.7) by (t − T0)

4 and set Zu,u(t) := (t − T0)4(Iu,u(t), 1)L2(�). Then, we have

∂tZu,u(t) + θ(t − T0)4‖∇xv(t)‖2

L2(�) � 4((t − T0)3, Iu,u(t))L2(�)

+ (|∂tu(t)|, (t − T0)4|Ju,u(t)|)L2(�) + (t − T0)

4‖B(u(t)) − B(u(t))‖2L1(�). (3.18)

Using (3.8)3, we can estimate the first term in the right-hand side of (3.18) as follows:

((t − T0)3, Iu,u)L2(�) � C(|B(u) − B(u)|1/2, (t − T0)

3|v|3/2)L2(�)

� C1‖B(u) − B(u)‖1/2L1(�)

(t − T0)3‖v‖3/2

L3(�)

� C2‖B(u) − B(u)‖2L1(�) + θ/4(t − T0)

4‖∇xv‖2L2(�). (3.19)

Analogously, using (3.8)2 and the embedding W 1,2(�) ⊂ L6(�), we can estimate the secondterm in the right-hand side of (3.18),

(|∂tu|, (t − T0)4|Ju,u|)L2(�) � C(|∂tu|, (t − T0)

4I1/2u,u · |v|)L2(�)

� C‖∂tu‖L6(�)[(t − T0)4(Iu,u, 1)L2(�)]

1/2(t − T0)2‖v‖L3(�)

� C‖∂t∇xu‖2L2(�)Zu,u + θ/4(t − T0)

4‖∇xv‖2L2(�). (3.20)

Inserting these estimates into (3.18), we finally have

∂tZu,u(t) − C‖∂t∇xu(t)‖2L2(�)Zu,u(t) + θ/2(t − T0)

4‖∇xv(t)‖2L2(�)

� C ′(1 + (t − T0)4)‖B(u(t)) − B(u(t))‖2

L1(�). (3.21)

Finite-dimensionality of attractors 1789

We recall that, due to our assumptions, the time interval [T0, T0 + τ ] is a regular interval withrespect to the solution u, i.e. on this interval, proposition 1.1 allows one to control the L2-normof ∂t∇xu, ∫ T0+µ

T0

‖∂t∇xu(t)‖2L2(�) dt � Q(‖u‖L∞([0,T ]×�)). (3.22)

Moreover, according to estimate (1.28), we have

‖B(u(t)) − B(u(t))‖L1(�) � CeK(t−T0)‖B(u(T0)) − B(u(T0))‖L1(�). (3.23)

Applying the Gronwall inequality to (3.21) and using (3.22), (3.23) and the fact that Zu,u(T0) =0, we deduce that

Zu,u(t) � Q(‖u‖L∞([0,T ]×�) + ‖u‖L∞([0,T ]×�))‖B(u(T0)) − B(u(T0))‖2L1(�),

t ∈ [T0, T0 + µ], (3.24)

for some monotonic function Q which is independent of the concrete choice of u and u.Integrating now inequality (3.21) with respect to t ∈ [T0 +µ/2, T0 +µ] and using (3.22)–(3.24),we obtain estimate (3.1) for the L2(W 1,2)-norm of v and finish the proof of proposition 3.1.

The next corollary is crucial in order to verify the second assumption of proposition 2.1in our situation.

Corollary 3.1. Let the assumptions of proposition 3.1 hold and let u, u and [T0, T0 +µ] be thesame as in that proposition. Then, the following estimates hold:

‖∂tB(u) − ∂tB(u)‖L2([T0+µ/2,T0+µ],W−1,2(�)) + ‖B(u) − B(u)‖L2([T0+µ/2,T0+µ],W 1,2(�))

� K‖B(u) − B(u)‖L1([T0,T0+µ/2]×�), (3.25)

where the constant K only depends on µ and the L∞-norms of u and u, but is independent ofthe concrete choice of u and u.

Proof. We first note that, for every two solutions u and u of equation (1.1) and every δ > 0,the following estimate holds:

‖B(u(δ)) − B(u(δ))‖L1(�) � Cδ‖B(u) − B(u)‖L1([0,δ]×�), (3.26)

where the constant Cδ only depends on δ and the L∞-norms of u and u. Indeed, in order toprove this estimate, it suffices to multiply equation (3.2) by tsgn(u(t) − u(t)), integrate over[T0, T0 + µ/2] × � and use the Kato inequality.

Combining the smoothing property (3.26) with proposition 3.1, we check that the followingestimate holds:∫ T0+µ

T0+µ/2‖u(t) − u(t)‖2

W 1,2(�) dt � C‖B(u) − B(u)‖2L1([T0,T0+µ/2]×�). (3.27)

In order to deduce (3.25) from (3.27), it is sufficient to note that, expressing ∂t (B(u) − B(u))

from equation (3.2) and using the fact that the L∞-norms of u and u can be controlled, wehave

‖∂t (B(u(t)) − B(u(t)))‖2W−1,2(�)

+‖∇x(B(u(t)) − B(u(t)))‖2L2(�) � C‖u(t) − u(t)‖2

W 1,2(�), (3.28)

where the constant C only depends on the L∞-norms of u and u. Thus, corollary 3.1 is proven.

We are now ready to finish the proof of theorem 2.2 by verifying the assumptions ofproposition 2.1 for some suitable discrete semigroup associated with equation (1.1). In order

1790 A Miranville and S Zelik

to construct it, we first construct a semi-invariant absorbing set B for the semigroup S(t)

associated with equation (1.1). As shown in the proof of theorem 2.1, the ball

B0 := {b0 ∈ L∞(�) ∩ W 1,2(�), ‖b0‖L∞(�) + ‖b0‖W 1,2(�) � R} (3.29)

is an absorbing set for this semigroup if R is large enough, but it is not necessarily semi-invariant. In order to overcome this difficulty, we transform this set in the followingstandard way:

B = [ ∪t�0 S(t)B0]L1(�)

, (3.30)

where [·]V denotes the closure in the space V . Then, on the one hand, this new absorbing setremains bounded in L∞(�)∩W 1,2(�) (due to theorem 1.1 and lemmas 1.2–1.3), i.e. for everytrajectory u(t) starting from B(u(0)) = b0 ∈ B,

‖u(t)‖L∞(�) + ‖u(t)‖W 1,2(�) + ‖∂tB(u)‖L2([t,t+1]×�) � C, (3.31)

where the constant C is independent of u and t � 0. On the other hand, this set is, obviously,semi-invariant with respect to S(t),

S(t)B ⊂ B. (3.32)

Then, according to proposition 1.1, there exists τ > 0 such that, for every trajectory u(t)

starting from B and every time interval [T , T + 1] of length one, there exists a subinterval[T0, T0 + τ ] ⊂ [T , T + 1] of length τ on which the L2-norm of ∂t∇xu is controlled as follows:∫ T0+τ

T0

‖∂t∇xu(t)‖2L2(�) dt � C, (3.33)

where C is independent of the trajectory u starting from B. Thus, it is sufficient to constructthe required exponential attractor on the absorbing set B only.

Let us now fix µ = 1/N , where N ∈ N is large enough so that

1/N � τ/3, (3.34)

and introduce the following spaces of functions depending on x and t :

H := L1([0, µ/2] × �),

H1 := W 1,2([0, µ/2], W−1,2(�)) ∩ L2([0, µ/2], W 1,2(�)).(3.35)

Then, H1 is compactly embedded into H (see, e.g. [LSU]). We also introduce the trajectoryanalogue of the absorbing set B as follows:

Btr := {B(u(t)), t ∈ [0, µ/2], u(t) solves (1.1) with B(u(0)) ∈ B} ⊂ H (3.36)

and define the µ/2-shift map S on Btr by

(Sv)(t) := S(µ/2)v(t), v ∈ Btr. (3.37)

Then, the semi-invariance (3.32) implies that the set Btr is also semi-invariant with respect tothe shift map S,

S : Btr → Btr. (3.38)

Our next task is to verify the conditions of proposition 2.1 for the map (3.38). Indeed, estimate(1.28) immediately implies that the map S is globally Lipschitz continuous on Btr and we onlyneed to verify the second assumption of proposition 2.1 and inequality (2.9). Indeed, due toour choice of the number µ, for every trajectory u(t) starting from B (or, equivalently, forevery trajectory of the discrete semigroup S(n) starting from b0 := {B(u(t)), t ∈ [0, µ/2]}),at least one of the intervals

[0, µ], [µ, 2µ], . . . , [(N − 1)µ, Nµ] (3.39)

Finite-dimensionality of attractors 1791

(let it be the interval [n0µ, (n0 + 1)µ]) belongs to the regularity interval of u, i.e. estimate(3.33) is satisfied on [n0µ, (n0 + 1)µ]. Thus, due to corollary 3.1, we have

‖Sw − Sv‖H1 � K‖w − v‖H, ∀v ∈ Btr, (3.40)

where w = S(2n0)b0. So, the second assumption of proposition 2.1 holds for S with N0 = 2N .Thus, we have proven that the discrete semigroup S(n) generated by the iterations of the

shift operator S possesses an exponential attractor Mtr ⊂ Btr which is finite-dimensional andsatisfies properties (1)–(4) of proposition 2.1.

We now recall that, due to estimate (3.26), the projection map ,

: Btr → B, v = v(µ/2), v ∈ Btr, (3.41)

is globally Lipschitz continuous. Consequently, projecting the trajectory attractor Mtr onto B,we obtain an exponential attractor Md := Mtr for the discrete semigroup {S(nµ/2), n ∈ N}on B which satisfies all properties (1)–(4) of proposition 2.1 with H = L1(�).

Thus, there only remains to pass from the exponential attractor Md of the semigroupS(nµ/2) with discrete times n ∈ N to the semigroup S(t) with continuous times t ∈ R+. Tothis end, we note that the map (t, b0) �→ S(t)b0 is uniformly Holder continuous with respect to(t, b0) ∈ [0, µ/2] × B with Holder exponent 1/2. Indeed, the Holder (and even the Lipschitz)continuity of S(t)b0 with respect to b0 is an immediate consequence of (1.28) and the Holdercontinuity with respect to t follows from the following simple estimates:

‖B(u(t + s)) − B(u(t))‖L1(�) = ‖∫ t+s

t

B ′(u(κ))∂tu(κ) dκ‖L1(�)

�∫ t+s

t

(B ′(u(κ)), |∂tu(κ)|)L2(�) dκ �(∫ t+s

t

B ′(u(κ)) dκ

)1/2

×(∫ µ/2

0(B ′(u(κ))∂tu(κ), ∂tu(κ))L2(�) dκ

)1/2

� Cs1/2. (3.42)

Thus, the required exponential attractor M for continuous times can be constructed by thefollowing standard formula:

M := [ ∪t∈[0,µ/2] S(t)Md

]L1(�)

⊂ B, (3.43)

see [EFNT] for more details. So, our main theorem on the existence of an exponential attractorfor the degenerate equation (1.1) is proven.

4. Generalizations and concluding remarks

In this concluding section, we discuss possible generalizations of the results obtained aboveand indicate several alternative methods to prove the finite-dimensionality of attractors.

Remark 4.1. To start with, we note that assumption (1.2)2 requires the nonlinearity a(∇xu)

to have a linear growth. However, this assumption is not essential and can be replaced by astandard polynomial growth of order p:

κ1(1 + |z|p−2) � A′′(z) � κ2(1 + |z|p−2), (4.1)

for some fixed p � 2 and positive constants κ1 and κ2. In that case, of course, we will, thanks toenergy inequalities, control the W 1,p-norm of the solution u (instead of the usual W 1,2-norm).Indeed, an accurate analysis shows that the global boundedness of A′′(u) has been used onlyin the proof of corollary 3.1 and only in order to obtain the control of the W−1,2-norm of

1792 A Miranville and S Zelik

∂t (B(u)−B(u)), see estimate (3.28). In the general case p > 2, this estimate fails and shouldbe replaced by an appropriate estimate of the Lq-norm, with 1

p+ 1

q= 1,

‖∂t (B(u) − B(u))‖2Lq([S,T ],W−1,q (�)) � ‖a(∇xu) − a(∇xu)‖2

Lq([S,T ]�)

+‖f (u) − f (u)‖2Lq([S,T ]×�) � C(‖u‖Lp([S,T ],W 1,p(�)) + ‖u‖Lp([S,T ],W 1,p(�)))

p−2

×(a(∇xu) − a(∇xu), ∇xu − ∇xu)L2([S,T ]×�) + C‖u − u‖2Lq([S,T ]×�)

� C1(1 + |T − S|p−2)(a(∇xu) − a(∇xu), ∇xu − ∇xu)L2([S,T ]×�), (4.2)

where we have used estimates (4.1), together with the fact that the L∞(W 1,p)-norms of u andu can be controlled. There remains to note that the scalar product in the right-hand side of (4.2)can be controlled by an analogue of proposition 3.1, see estimate (3.3). Thus, the estimateof corollary 3.1 remains true if we replace the L2(W−1,2)-norm by the Lq(W−1,q)-norm and,consequently, the space H1 in (3.35) should be replaced by

H1 := W 1,q([0, µ/2], W−1,q(�)) ∩ L2([0, µ/2], W 1,2(�)).

Since this change does not destroy the compactness of the embedding H1 ⊂ H, the remainingof the proof of theorem 2.2 does not change as well. Therefore, the main result of this paper(theorem 2.2 on the existence of a finite-dimensional exponential attractor) remains true underthe more general assumption (4.1).

Remark 4.2. We now discuss the regularity assumptions of the domain �. Indeed, althoughwe have assumed the boundary ∂� to be smooth, this assumption has been used only inlemma 1.3 (in order to verify the L2(W 2,2)-regularity of the solutions) and in theorem 1.2 (inorder to make sure that the solutions un of the approximate problem (1.6) are regular enough).However, the W 2,2-regularity of the solutions is, in fact, nowhere used in the sequel and all theother estimates do not require the domain � to be regular. Indeed, we have factually only usedthe Sobolev embedding W

1,20 (�) ⊂ L6(�) and some interpolation inequalities which do not

require any regularity of the boundary (due to our choice of Dirichlet boundary conditions; forNeumann boundary conditions, the Lipschitz continuity of the boundary is required). Thus,analysing the solutions of the approximate problems (1.6) in a more accurate way, we see thatthe main results of the paper hold, e.g. for Lipschitz domains (and even for some non-Lipschitzones).

Remark 4.3. We now note that the above results are also valid (with a lot of simplifications)in the non-degenerate case

B ′(u) � κ > 0 (4.3)

as well. Indeed, in that case, assumption (1.44) is automatically satisfied (for B of class C2)and assumption (1.4) also holds automatically, since, now, B(z) ∼ z near zero. However, ourmethod may seem artificial in this situation. Indeed, under assumption (4.3), equation (1.1)can be rewritten in the form of a quasilinear second-order parabolic equation,

∂tu = [B ′(u)]−1A′′ij (∇xu)∂xi

∂xju − [B ′(u)]−1(f (u) − g). (4.4)

The analytic properties of such equations in the non-degenerate case are very well understood,see, e.g. [LSU], and we can use the classical and powerful regularity theory of such equations.Indeed, in particular, if B, a ∈ C2(R) and g ∈ Cα(�), for some α > 0, then, due to the interiorregularity estimates, equation (4.4) (or, equivalently, equation (1.1)) possesses an absorbingball in the space C1+α/2,2+α([T , T + 1] × �),

‖u‖C1+α/2,2+α([T +1,T +2]×�) � Q(‖u(T )‖L∞(�)), (4.5)

Finite-dimensionality of attractors 1793

see, e.g. [LSU], chapter 6, sections 1–6. Having this estimate, one can verify the finite-dimensionality of the global attractor, e.g. by the classical volume contraction method andverify the existence of an exponential attractor by proving the following simpler smoothingproperty for the difference of two solutions:

‖S(t)b10 − S(t)b2

0‖H 1(�) � CeKt

√t‖b1

0 − b20‖L2(�), t > 0, (4.6)

instead of the complicated version of such an inequality formulated in proposition 2.1. Thus,our paper is mainly oriented towards the degenerate case when (4.3) fails and when the reductionto (4.4) and the regularity (4.5) also fail (see also the next remark).

Remark 4.4. We now discuss an alternative method to prove the finite-dimensionality of theglobal attractor in the degenerate case (when assumption (4.3) is not satisfied). In contrast tothe regular case, one cannot expect the existence of classical solutions or/and the smoothingproperty 4.5 to hold in the degenerate case (even if all the terms are of class C∞) and the bestregularity which can be expected for our equation is the following Holder continuity:

‖u‖Cα/2,α([T +1,T +2]×�) � Q(‖u(T )‖L∞(�)), T > 0, α > 0, (4.7)

see [D2, DUV] and the references therein for precise conditions which guarantee Holdercontinuity results for degenerate second-order parabolic equations.

However, there exists a general method (suggested in [EfZ1]) which allows one to extractthe finite-dimensionality (and the existence of an exponential attractor) from this Holdercontinuity and the L1-Lipschitz continuity with respect to the initial data. The applicationof this method to our problem gives the following result.

Proposition 4.1. Let the assumptions of theorem 1.1 hold and let, in addition, a, B be ofclass C2, the Holder continuity estimate (4.7) be satisfied and the following monotonicityassumption:

|f (z1) − f (z2)| � κ|B(z1) − B(z2)|, κ > 0, (4.8)

hold, for every z1 and z2 in a small neighbourhood of all the degeneration points of B. Assumealso that g ∈ Cα(�), for some α > 0. Then, the global attractor A of problem 1.1 isfinite-dimensional and there exists an exponential attractor for this problem in the sense ofdefinition 2.2.

Sketch of the proof. We briefly recall here the main idea of [EfZ1] by considering, forsimplicity, the case of one degeneration point for B at z = 0, i.e. assumptions (A) hold. Inthat case, we can consider the usual framework, i.e. S(t) maps u(0) onto u(t). Let B be anabsorbing set of the semigroup S(t) (for which the uniform Holder continuity holds due to(4.7)) and let Bε(u0) be an ε-ball in the metric of L1(�) centred at u0 ∈ B. Then, since u0 iscontinuous, we can split the domain � into the union of two subdomains,

� = �+(u0) ∪ �−(u0), �+ := {x ∈ �, |u0(x)| > β},�− := {x ∈ �, |u0(x)| < 2β}, (4.9)

where β is a sufficiently small positive number. Moreover, since the semigroup S(t) is globallyLipschitz continuous in the L1-metric and the norm ‖u‖Cα/2([0,T ]×�) is uniformly bounded withrespect to all trajectories starting fromB, then, for ε sufficiently small, there existsT > 0 (which

1794 A Miranville and S Zelik

is independent of u0 and ε) such that, for every trajectory u(t) such that u(0) ∈ Bε(u0) ∩ B,the following estimates hold:{|u(t, x)| > β/2, (t, x) ∈ [0, T ] × �+,

|u(t, x)| < 3β, (t, x) ∈ [0, T ] × �−.(4.10)

In other words, all the trajectories starting from Bε(u0) ∩ B remain uniformly close to thedegeneration point z = 0 for x ∈ �− and are uniformly non-degenerate for x ∈ �+ if t � T .This simple observation is the key point of the method and is the precise reason why we needan assumption on the Holder continuity.

Furthermore, since the above trajectories are (uniformly) non-degenerate on [0, T ] × �+,then, (4.3) holds on this domain and we can rewrite equation (1.1) in the form (4.4)on the domain �+. Then, the parabolic regularity theorem mentioned above yields thatu ∈ C1+α/2,2+α([0, T ]×�+) and the norm in this space is uniform with respect to all trajectoriesstarting from Bε(u0) ∩ B. Therefore, the difference v(t) := u1(t) − u2(t) between twosuch solutions restricted to �+ satisfies a linear second-order parabolic equation with regularcoefficients and, thus (roughly speaking, see [EfZ1] for a precise formulation), we have, forthe �+-component of the difference v(t), a smoothing property which is analogous to (4.6) (ofcourse, v

∣∣∂�+

�= 0, so that the precise formulation should contain some interior estimates andcut-off functions).

On the other hand, fixing β > 0 small enough so that inequality (4.8) holds, for all z1 andz2 with |zi | � 3β, i = 1, 2, and applying the L1-Lipschitz continuity estimate in the domain�− to the equation for the difference v(t), we have

‖B(u1(t)) − B(u2(t))‖L1(�−) � e−κt‖B(u1(0)) − B(u2(0))‖L1(�−), t � T (4.11)

(again, roughly speaking, since v∣∣∂�−

�= 0, and some cut-off functions are necessary).Thus, we have a uniform (with respect to all initial data belonging to Bε(u0)∩B) contraction

of the �−-component of the difference of two solutions and a uniform smoothing propertyfor their �+-component. As shown in [EfZ1], this decomposition is sufficient to verify thatthe image S(T )(Bε(u0) ∩ B) of an ε-ball centered at u0 can be covered by a finite number N

of γ ε-balls in L1(�), with γ < 1 and N independent of ε and u0, and this last assumptionimplies the finite-dimensionality and the existence of an exponential attractor, see [EfZ1] fordetails. Thus, proposition 4.1 is proven.

The essential advantage of the method introduced above is that, in contrast to our schemedeveloped in sections 2 and 3, where we indeed need the nonlinearity B to be of class C5 nearthe degeneration points, this method does not require any regularity on B near the degenerationpoints and B of class C2 is necessary only outside the degeneration points (i.e. in the domain�+). Near the degeneration points (i.e. in �−), we do not need any regularity assumption onB and only need the monotonicity estimate (4.8) to be satisfied.

However, in contrast to the classical De Giorgi theory for non-degenerate second-orderparabolic equations, the Holder continuity for degenerate equations is a non-trivial and delicatefact which can be even violated, e.g. for some elliptic–parabolic equations. In fact, we do notknow whether or not the Holder continuity holds under assumptions (B) in the elliptic–paraboliccase. That is the reason why we chose to give an alternative proof which is based on relativelysimple energy type estimates in this paper.

Remark 4.5. To conclude, we note that it would be very interesting to prove the existenceof exponential attractors when the function B(u) has singularities or discontinuities (e.g.for Stefan-like problems). However, the above methods do not work in this situation, sinceinequalities of the form (4.8) cannot be satisfied if B has singularities and f is regular. The onlytype of singularities which we are able to treat are the discontinuities of the derivatives of B.

Finite-dimensionality of attractors 1795

Indeed, let us consider the following particular case of equation (1.1):{∂tB(u) = a�xu − f (u) + g,

u∣∣∂�

= 0, B(u)∣∣t=0 = b0,

(4.12)

where a > 0 is some fixed number and the function B is only Lipschitz continuous,

κ1|z1 − z2|2 � [B(z1) − B(z2)].(z1 − z2) � κ2|z1 − z2|2, (4.13)

with positive constants κ1 and κ2.

Proposition 4.2. Let the nonlinearity B satisfy (4.13) and the nonlinearity f be Lipschitz andsatisfy the dissipativity assumption (1.3). Then, the semigroup S(t) associated with equation(4.12) via (2.2) possesses an exponential attractor M in the sense of definition 2.2.

Sketch of the proof. Let u1(t) and u2(t) be two solutions of (4.12) starting from an absorbingset in L∞(�). Then, multiplying the equation for the difference of two solutions u1 and u2

by (−�x)−1(B(u1) − B(u2)) and using (4.13), together with the Lipschitz continuity of f ,

we haved

dt‖B(u1) − B(u2)‖2

H−1(�)

+ θ‖B(u1) − B(u2)‖2L2(�) � C‖B(u1) − B(u2)‖2

H−1(�), (4.14)

with some positive constants C and θ which are independent of u1 and u2. Fixing now someT > 0, multiplying (4.14) by t and integrating over [0, T ], we deduce that

‖B(u1(T )) − B(u2(T ))‖2H−1(�) � CT ‖B(u1) − B(u2)‖2

L2([0,T ],H−1(�)). (4.15)

Integrating then relation (4.14) with respect to t ∈ [T , 2T ] and using (4.15), we infer

‖B(u1) − B(u2)‖L2([T ,2T ],L2(�)) � CT ‖B(u1) − B(u2)‖L2([0,T ],H−1(�)), (4.16)

with some constant CT depending on T . Finally, we deduce, from the equation for thedifference between u1 and u2, together with (4.13) and the Lipschitz continuity of f , that

‖∂tB(u1) − ∂tB(u2)‖L2([T ,2T ],H−2(�)) � C‖B(u1) − B(u2)‖L2([T ,2T ],L2(�)). (4.17)

Thus, combining (4.16) and (4.17), we have

‖B(u1) − B(u2)‖W 1,2([T ,2T ],H−2(�))∩L2([T ,2T ],L2(�))

� CT ‖B(u1) − B(u2)‖L2([0,T ],H−1(�)), (4.18)

with some positive constant CT which is independent of the choice of the trajectories u1 andu2 starting from the absorbing set.

Since the embedding

W 1,2([0, T ], H−2(�)) ∩ L2([0, T ], L2(�)) ⊂ L2([0, T ], H−1(�))

is compact, inequality (4.18) allows indeed to construct an exponential attractor for thesemigroup S(t) in the topology of H−1(�) by the l-trajectories method, see [MP]; see also [Ro].Since the H 1-norm can be controlled on the absorbing set, we obtain, by interpolating betweenH−1(�) and H 1(�), the existence of an exponential attractor in the topology of L2(�) (andeven Lp(�), p finite) as well and proposition 4.2 is proven.

Example 4.1. Although the result of proposition 4.2 does not apply to the Stefan problem,it gives however the finite-dimensionality of attractors for some free boundary problems. Inparticular, the following free boundary problem:

∂tu = a1�xu − f1(u), u > 0,

∂tu = a2�xu − f2(u), u < 0,

u∣∣+ = u

∣∣−, ∂nu

∣∣+ + ∂nu

∣∣− = 0, u = 0,

(4.19)

1796 A Miranville and S Zelik

where a1, a2 > 0, the functions fi, i = 1, 2, satisfy the dissipativity assumption (1.3) andf1(0) = f2(0) = 0, can be rewritten in the form (4.12), with

B(z) ={a−1

1 z, z > 0,

a−12 z, z < 0,

f (z) ={a−1

1 f1(z), z > 0,

a−12 f2(z), z < 0.

Thus, all the assumptions of proposition 4.2 are satisfied and, consequently, problem (4.19)possesses a finite-dimensional exponential attractor in the phase space L2(�).

Acknowledgment

The authors wish to thank Professor V Vespri for very useful discussions. They also wish tothank the referees for their careful reading of the paper and for useful comments.

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