Pergamon

Nonlinear Analysis, Theory, Methods&Applications, Vol. 23, No. 9. pp. 1189-1209. 1994 Copyright 0 1994 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0362-546X/94 S7.CQ+ .CKl

THE DEBYE SYSTEM: EXISTENCE AND LARGE TIME BEHAVIOR OF SOLUTIONS

PIOTR BILER, WALDEMAR HEBISCH and TADEUSZ NADZIEJA Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland

(Received 15 March 1993; received for publication 20 December 1993)

Key words andphrases: Parabolic-elliptic system, nonlinear boundary conditions, existence of solutions, asymptotic behavior.

1. INTRODUCTION

The aim of this paper is to study the Debye system from the theory of electrolytes (see, e.g. [l-3]) which consists of two parabolic equations

u, = V*(Vu - UVV) (1)

V, = V*(Vu + VVP) (2)

coupled through the electric potential v, satisfying

Ay,=u-v (3)

in a bounded domain Sz of /R” with smooth boundary &2. The natural (no-flux) boundary conditions are

au -- $2=* av av

av a@ ~+?I~=0 (5)

and

v=o (6)

where v denotes the unit normal vector to aa. This system, supplemented with the initial conditions

U(X, 0) = u&x) 1 0 (7)

u(x, 0) = trg(x) 1 0, (8)

and &x, 0) determined by the relations (3) and (6), describes the evolution of the densities u and v of ions in an electrolyte. Besides electrochemistry, such systems appear in the theory of semiconductors, cf. e.g. [4-81, where u and u also describe densities of charge carriers: electrons and holes.

Note that the drift coefficient in both equations (I), (2) can be reconstructed from (3) and (6), which leads to a system of differential-integral parabolic equations of Fokker-Planck type. However, this interpretation does not provide a substantial simplification of the problem.

1189

1190 P. BILER et al.

The flux conservative form of equations (l), (2) shows that the boundary conditions (4), (5) are the simplest ones guaranteeing the conservation of the total charges of both kinds of ions: SnU(X,t)~= Sn&l(x)d.G and Jn v(x, t) dx = 1 e u,,(x) dx as long as solutions u, u exist (in a reasonable weak sense).

The nonlinear boundary conditions (4), (5) constitute the major difficulty in the mathematical analysis of the problem. When simpler boundary conditions (like the Dirichlet, the homogeneous Neumann, or a third type of linear boundary conditions) are imposed, see [4-81, the analysis of boundary integral terms in various a priori estimates becomes much easier than in the physically relevant case of no-flux conditions.

The one-dimensional version of the problem (l)-(8) with only one charge density u (i.e. with v = 0) has been studied in [3]. The existence of global classical solutions and their convergence to stationary solutions has been proved using a generalization of the Hopf-Cole change of variables transforming equation (1) into a linear heat equation.

The paper [9] has also dealt with a simplified model corresponding to the case u = 0 in (l)-(8). The well-posedness locally in time of the initial-boundary value problem has been proved in the three-dimensional case. Moreover, in the global existence case, the convergence of solutions to (uniquely determined by the initial data) stationary states has been established. However, the problem of global in time existence of solutions remained an open problem except for the cases of the initial data close to steady states (n = 2,3) or small initial data (n = 2).

The full Debye system considered here with two parabolic equations coupled through the potential v, is more difficult to study than a single parabolic one as in [9]. Using a similar method as in [9], we prove, in this paper, local well-posedness of the problem (l)-(8) in L2(SJ) for n = 2,3 (theorem l), and in P(n), p > n (theorem 2). Then, we show the global in time existence of solutions with a suitable uniform boundedness estimate in two-dimensional domains for arbitrarily large initial data (theorem 3), as well as for the balls in IR” (the radial case, theorem 5). We note the positivity preserving character and the conservation of total charge properties of weak solutions of (l), (2) (proposition 1). Local regularity of solutions is studied in theorem 2. Under the assumption of global existence we prove the convergence of solutions to the stationary states (theorem 6) whose structure has been studied in [lo] (cf. proposition 2). Liapunov functions (proposition 3) play an important role in establishing such asymptotic results. The stationary solutions have open basins of attraction, i.e. initial data close enough to them develop solutions converging to these attractors (theorem 4).

Here, it is worth noting that replacing equation (1) by u, = V - (Vu + u VP) and putting v = 0 we obtain a model of gravitational interaction of particles. The resulting system has stationary solutions for total mass Jn u(x, t) dx = Jn u,,(x) dx small enough only, and it may develop singularities of u (blowing up solutions) in finite time. Related questions are the topic of current studies, cf. [ll].

In the sequel we use standard notation IulP for the LP(Q) norms of functions, and [lulls for the HS(Q) norms. The constants independent of functions defined on Sz will be denoted by the same letter C, even if they may vary from line to line. Tr.e letters M, N will be used for quantities whose dependence on initial data should be controlled in the proofs. We refer to [12] for various Sobolev imbeddings, traces and interpolation inequalities.

Some parts of the proofs in this paper will be merely sketched. In such a situation [9] is a standard reference, where the details of reasoning for the simplified problem may be found. Adapting them to the full Debye system is a quite easy task.

Debye system 1191

2. LOCAL EXISTENCE OF SOLUTIONS

Similarly, as in [9] weak solutions of (l)-(8) are understood as the weak solutions of equations (l), (2) with Vy, considered as a known coefficient which is related to (1) and (2) via (3). In such an interpretation (l), (2) become linear uncoupled equations, so the theory in [13, Chapter III] applies, and we should only bother about equality (3). Finally, the solutions are furnished by a fixed point argument founded upon a compactness property of parabolic equations. We begin with the (two- and) three-dimensional case.

THEOREM 1. Consider a bounded domain G in R3 with Cl+& boundary ati for some E > 0. Given arbitrary initial conditions (7), (8) u,, , v, E L’(sZ) there exists T = T(lu&, ]~+,1~) such that the problem (l)-(8) has a weak solution (u, v> with u and u belonging to the space V, = L”((0, T); L2(sZ)) n L2((0, T); H’(C2)). Moreover, u,, u, E L2((0, T); H-‘(i-2)).

Proof. The space in the conclusion of theorem 1 is just the standard space V2 = V,(G x (0, T)) used in the theory of weak solutions of linear parabolic equations in divergence form in [13]. We say that the couple (u, v) with U, v E V2 is a weak solution of the problem (l)-(8) on Sz x (0, T) if for each couple (q, [> of test functions q, [ E H’(fi x (0, T)) the following integral identities hold for a.e. t E (0, T)

and for a.e. t E (0, T) a, = cp(+, t) is a weak solution of (3), (6), i.e.

This is a modification of standard definitions of weak solutions of initial-boundary value problems from [13, Chapter III, Sections 1, 4, 51 taking into account the no-flux conditions

(4)s (5). Such a solution satisfies the energy (in)equalities

3 s t u2(x, t) dx + s.T (Vu - 24 Vfp) - vu = 3 z&x) dx,

n 0 0 s n

+ s t u2(x, t) dx +

n IS (Vu + u Vq?) - vv = + 0 n s v,“(x) dx

n

valid for all t E [0, T]. This is proved for a.e. t E (0, T) observing that

ut, ut E L2W, T); ~-‘(W

(see the last part of the proof of theorem 1) and the fact

u, u E L2((0, T); N’(i-2)).

1192 P. BILER et al.

Then invoking [13, Chapter III, theorem 5.11 gives the continuity U, v E C([O, T];L’(Q)), hence, the above energy relations follow. By abuse of language we will write them in the differential form

1 d j $ IUI; + pv1; = -

s vvv - vp, (10)

n

which is formally obtained by taking the L* inner product of (1) and (2) with U, U, respectively, and integrating by parts. In the sequel we will be performing some formal calculations leading to differential inequalities, but we will understand them in the proper integral formulation like the energy relations above.

Define for a fixed T > 0 ‘X as the space of vector valued functions

l<u, u>: u, u E L4((0, T); L%-J))l

whose norm

Ill<% u>III = IMII + lll4ll~ llblll = ({;( in l W, 01’ hy df>“4

is finite. For each element (ii, 0) of ‘X the weak solution cp = ~(x, t) of the Poisson equation Ap = ii - fi with the boundary condition (6) is defined for a.e. t E [0, T]. It follows from the Sobolev theorem (for n 5 3) that the potential p, denoted symbolically a, = A_d(ii - p), A, being the Dirichlet Laplacian in Sz, satisfies Vcp( *, t) E L6(C2), since ii( * , t) - ii( *, t) E L2(Q) for a.e. t E [0, T]. In the sequel we will not write explicitly the dependence of the considered functions on t, so we note

Iv& 5 c(ii - U12.

Observe that Vy, = V(A_d(ii - u)) depends measurably on t. Moreover, we have

Vq E L4((0, T); L’(Q))

since lr(je Ivp(t)\6)2’3 dt 5 clllii - ~(11~.

(11)

From the solvability conditions for the linear equations in [13, Chapter III, Sections 4, 51 we infer that the system (1) and (2), with cp determined by (ti, U>, boundary conditions (4), (5) and the initial conditions (7), (8), does have a solution (u, v> on [0, T]. Indeed, the coefficient Vq in the equation should satisfy the relation IVv12 E L’((0, T); Lq(sZ)) with l/r + n/2q I 1, and the coefficient @,/av in the third type boundary condition should be in Lr’((O, T); Lq’(%2)) with l/r’ + (n - 1)/2q’ 5 l/2. This is verified in our case since n = 3, q = 3, r = 2; n - 1 = 2, q’ = 4, r’ = 4 (for the Sobolev imbedding theorem for traces of functions, here N’(Q) c L4@Q), we recall [12, theorem 5.221).

The energy (in)equalities for the auxiliary linear problem just solved, cf. [13, Chapter III, Section 21, assume the form

(13:

Debye system 1193

The right-hand sides of (12) and (13) are now estimated by

luvu * vpl, I lvu~zlul&yol~ 5 clvul~llull:‘21Ult’21ii - VI2 5 ~1124: + cl&b - 4;

and IVVU - vqli I $llr# + c[v/~~u - uI~, respectively. Summing up (12) and (13) we arrive at the inequality

‘; 10.4, Id; + Iv04 & 5 (Clii - 4:: + I)l(u, VA; =: c.uwl<u, u)l22 (14)

where [(u, u>Iz = 1~1; + Ivli, and a(t) is a function of t integrable over (0, T). The above inequality leads to

(15)

or Ilk4 u)lIl 5 T”41( uo, 14,))~ exp(i ]~a@) ds). This means that for sufficiently small T > 0 and R large enough the ball {(ii, ti> E X: IIJ(ii, ~>I11 I R) is invariant under the nonlinear operator 3t: ‘X 4 X defined by ‘X((ii, 0)) = (u, u). This property, as well as the continuity of 3t in the norm Ill* 111 of the space EC, is proved exactly as in [9].

The next property of 3t: the closure C of the image X(((ii, a>: /[[(ii, ~>lll I R)) of each ball in X is compact in ‘X, is a consequence of the Aubin-Lions compactness lemma [14, Chapter I, Section 5.21. Namely, (15) implies U, u E L”((0, T); L2(L2)) C L2((0, T); L’(Q)), and then from (14) we infer U, u E L2((0, T); H’(L2)). We also gain some regularity in t, that is j,‘\\d/dt(u(t), u(r)>1& dt < co, which is proved by applying to equations (l), (2) any test function from H’(Q), cf. lemma 2 in [9]. Then e is compact in the L’((O, T); L’(Q)) norm by the Aubin-Lions lemma, and its local boundedness in the L”((0, T); L2(L2)) norm implies its compactness in ‘X = (L4((0, T); L2(sZ)))‘. Now, any fixed point of the operator 32. (which does exist by the Schauder theorem) solves the problem (l)-(8) in the weak sense on (0, T).

We note in the following proposition useful facts about (u, u) which can be proved in a standard manner along the lines of the proof of propositions 1 and 2 in [9].

PROPOSITION 1. If U,,(X) 1 0, u,,(x) zz 0, then U(X, t) r 0, u(x, t) 1 0 for a.e. x E Sz and t 2 0. Moreover, the total charges are conserved: lu(t)( 1 = In u(x, t) dx = jn z+,(x) dx = lu,,l 1,

Iwl, =SnW,O~= Sn%cd~= IdI.

Solutions with nonnegative initial conditions depend continuously on the initial data, so in particular (u, u) solving (l)-(8) in the weak sense is unique. More precisely, the triple (u(t), u(t), V&t)) depends continuously in the (L2(s2))3 norm on

(~0, uo, v&h, - uo))).

Remark 1. It is clear that the proof of theorem 1 reproduced from [9] applies to the gravitational case, when the first equation of the system is replaced by U, = V * (Vu + u Vy1)

and u = 0. Having in mind the remarks in the Introduction on the nonexistence of global solutions emanating from large initial data for this problem, we should note that any attempt to prove the global existence of solutions in the electrolytic case needs a more subtle analysis of the terms on the right-hand sides of (9) and (10).

1194 P. BILER et al.

In our construction, to have solutions global in time, it is sufficient to obtain an a priori estimate on the L’(n) norm of (u, u> locally uniform in time. We will show such an estimate in certain cases in the next section, but here we would like to point out that in the case of Sz c IR3 it suffices to have such an estimate only in Lp(sZ), p > 312. This follows from Amann’s theory of parabolic systems with nonlinear boundary conditions (a personal communication), but this can be proved using the analogs of (9), (10) obtained by taking the inner products of (l), (2) with up-‘, VP-‘, respectively, compare also the proof of theorem 2 below.

We recall here a part of remark 3 in [9] which explains the following remark.

Remark 2. A direct attempt to prove a global a priori bound in the L2(G) norm for (u, u) leads to the integral jn (U VU - u Vu) - Vcp, which (even after certain integration by parts) in the general case cannot be estimated crudely for n = 3 using the Sobolev-Gagliardo-Nirenberg inequalities. The reason is that the, so-called, net smoothness of the only terms l(u, u>li, IV(u, u)l: that could be used to compensate the above integral is strictly less than its net smoothness. The same can be verified for the term jan u2a(p/av obtained by the integration by parts of ja u VU - Vy, = -t ja u3 + t jan u2 ayl/av (here we supposed for simplicity u = 0). It cannot be compensated by IVul:, lul$, lul:, when it is majorized crudely. For n = 2 their net smoothness coefficients coincide. Exactly the same difficulty is met when we attempt to prove uniform boundedness in time of any Lp(Q) norm of u(t), p > 1. For the radial case additional information is available which allows us to prove an LP(Q), p > n, a priori bound uniform in time, cf. theorem 5 below.

Next we will study the n-dimensional case, n r 4, and local in time regularity of solutions of (l)-(3). Let us remark that solutions considered in theorem 1 for n = 2, 3 satisfy (iii), (iv) below.

THEOREM 2. Suppose that Q is a bounded domain in IR” with Cl+” boundary &2 for some & > 0.

(i) If p > n and 0 I uO, u,, E Lp(sZ), then there exists Tp = T(p, lu&,, It&,) such that the problem (l)-(8) has a weak solution (u, u) in the space (L”((0, T); Lp(s2)))2. Moreover, up’2 9 Up” E L2((0, T); H’(Q)) and such a solution is unique.

(ii) If p > n/2, 0 5 ue, u0 E Lp(Q), then there exists a local weak solution (u, u) of the problem (l)-(8) in (L”((0, T); Lp(s2)))2.

(iii) The solutions considered in (i) and those constructed in (ii) are locally bounded in time, in the sense that U, u E &((O, T); L”(CJ)).

(iv) Given two solutions (u, , u,) and (u,, u2) of (l)-(6), if

M = 1(%(O) - U,(O)> %(O) - ~2Pwlp

for some p > n is small enough, then for each 6 > 0 and f E (6, T)

I(W) - UZ(f), UlW - vzw>L 5 ww

with some constant C(6). Moreover, the solution (u, u) is locally Holder continuous in (x, t).

Debye system 1195

Proof. (i) We prove the local in time existence of solutions to (l)-(8) following remark 2 in [9]. The construction is entirely similar to that in theorem 1. Its applicability is based on the estimate (V& d C(n,p)lu - ulP (instead of (11) in the proof of theorem 1). Any solution obtained from the Schauder fixed point theorem is a weak solution in the sense of theorem 1, and this is unique in the class L”((0, T); E’(Q)). The estimate for the time of existence Tp obtained in [9] is mediocre (even worse than that in the L2 framework for n = 2,3). The regularity of solutions of (l)-(8) follows from (20) below. However, this can be improved; as we will see in (iii), such solutions are in L”(a) for a.e. t E (0, T).

(ii) Let us begin with a general calculation valid for weak solutions, when in the definition suitable cut-off functions of solutions will be taken as test functions.

Consider p > n/2 1 2, and any convex nondecreasing function v/ such that 0 I IJ/’ is compactly supported. Define the function x by x(O) = 0 and x’ = (I@‘)~‘~. Of course, w is a Lipschitz function of linear growth. From the regularity properties of weak solutions the relation d/dt I&) = Us@ makes sense, hence, we have

d

5, s u/(u) =

=

=

- s (Vu - u Vtp) * V(ly’(u)) n

- s JVuJ2yl”(u) + 24 vu - Vqy”(u) n s R

- s ~ lv(xw)12 + s 4W”w1’2 V(XW) * VP, R

(16)

and by the same reasons

Now assume that u(@‘(u))“~ inequality

; s *(v(u) + v(e) +

I px(u) and x(u) d CW(U)“~, which leads from (16) to the

IV(xo49 xw; 5 PlVcd~h XoJ)>I2I~XW~ xw>I,lwI, (17)

with any q > n and l/q + l/r = l/2. Since r < 2n/(n - 2) and by the Sobolev imbedding theorem H’(a) C Lzn’(n-Z)(Q), w e can estimate Ix(u)l, I Cl~(x(u))l:-“lx(u)l~ with some cx > 0 (in fact 01 = 1 - n/q). So the right-hand side of (17) is less than

CPIV(XW~ xw>l;-~lww29 v/w1’2~l;lw, 5 3lv<xo4, xwl; + CP21(XW, xw)l~Ivd~.

Consequently, from (17) we obtain in the integrated form for s I t

s (v(u)(t) + v(W)) + i

n s ’ I V(xW(r), x(M@> 1; dr s

I exp Cp2 ’ (VyI(r)[i’“l ( s

(w(u)(s) + JY(4(4). s n

(18)

1196 P. BILER et al.

Next, define wk by V/~(U) = u for u I 0, v;(u) = p(p - l)upm2 for 0 < u s k, and w;(u) = 0 for u > k. Using the inequality (18) with w replaced by v/~ we arrive in the limit k -+ 00 at the relation

5 (d’(r) + d’(t)) + ’ [V(U~‘~(T), u”“(7))l; dr 0 S

5 exp Cp2 ’ IVq(r)\ydr ( s >S

W(s) + fl(s)). (19) S n

This means that the norm I(u, u)J,, is controlled as long as the quantity 5: Iv&‘~ is finite. From the properties of weak solutions to the Poisson equation in Q (see [15, Chapter II, Sections 3, 41; here the assumption concerning a&J is used) we have a generalization of (11)

hl, 5 CobP)lu - VIP

with l/q = l/p - l/n; herep > n/2, hence, q > n. Choosing a sufficiently small T > 0, e.g. T = (3Cp2(C(n,p)J(u,, u,)),)~~‘(~~-“))-~ (remember that CY = 1 - n/q = (2p - n)/p) we obtain from (19) an estimate of <u(t), v(t)> in LP(n) independent of the actual value of IV&

on to, Tl

t~~PTIlwt)9 w>l, 5 3la409 h&’ SUP Iwt)lq 5 c. tE I0.U

Now, this is a standard limiting procedure which allows us to obtain solutions in (ii) by approximating them by solutions constructed in (i), with regular initial conditions approximating those from LP(n), p > n/2.

Indeed, the above bound on (u, u) makes possible the continuation of solutions with regular initial data in Lpo(sZ), p. > n, step by step to the whole interval [0, T], with a uniform bound in Lp(sZ), p > n/2. This, together with the compactness obtained from the integrability of IW2w, vp’2w>lt, allows the passage to the limit of regular solutions.

(iii) Now suppose that for somep > n/2 the norm I(u, u)lp is bounded on an interval (0, T,). The imbedding H’(Q) C L 2n’(n-2)(Q) makes possible the following recurrence argument: for a.e. 0 < t c q up’2(t), fl’2(t) E H’(SZ), so this implies u(t), u(t) E Lp”““-2)(n). Repeating the preceding reasoning gives uPn’(2(n-2))(t’), UPn’(2(n-2))(t’) E H’(Q) for t’ arbitrarily close to t, together with a uniform estimate. Consequently, the exponent of the LP-regularity improves each time by the factor n/(n - 2) > 1, from p > n/2 up to an arbitrary exponent p < 00 (for our purposes we stop when p > n).

For a sharper result on L”(Q) boundedness, beginning from a p. > n we can apply an even simpler and more powerful argument. Namely, suppose that the inequality

$ J(u, u,I”, + 4(pp l) lv(up’2, tF2>1;

5 2(p - 1)Iv(up’2, vp’2>121W’2, ~‘2)121vcDlco

~ 4(P-1)

( P - 1 >

(V(uP’2, ?Y’2>I$ + Mp21(zF2, vp’2>(;

Debye system 1197

for p 2 p. (derived like (19)) holds on some segment Z = [s, T], 0 I s < T, with a constant A4 = sup#& I csup,lu - ul& 5 CSUP,l(~, el;, < 00, since Iv~[_, I CIU - ul,,,, from the Sobolev imbedding theorem. The resulting inequality

-$ IN, u>I$ + IvW2, t9’2>l; 5 MP21(U, VA;,

after the integration on [s, t] with t I T, implies (cf. (19) after estimating Iv~I~,)

IWO, u~>lfd + s ’ IV(U~‘~(T), vP’2W1~ dr 5 exp(Mp2(t - s))ku(.+ u(s)>lfJ. (21) s

The inequality (21) is valid for p = po, and this allows us to start a recurrence argument leading to the finiteness of the L”(Q) norm of (u, u) for every r E (0, T]. With no loss of generality suppose that r = T, and take T - 1 I to < T, (T = n/(n - 2) for n 1 3, Q = 3 for n = 2, 6, = (T - &)/a. Define for m L 1 p,,, = pOom, d,,, = c500-2m, the segments Z, = [T - a&, , T - S,], and the sequence N, = supTE tt,, TIj(~(r), u(r)) IPm , where t, E Z, will be determined later. We would like to estimate N,,, by N,,, , and then by N, which is finite from part (i). From inequality (21) with s = t, E Z,, t = T - dm+l (so [s, t] 3 Zm+l), and p = pm we obtain

hence, for some t,+l E Zm+l (from the imbedding H’(Q) c L2”(S’Q)

lWL+,)9 a?I+I))l:;m = Iwm’2&n+d, ~m’2ct,+I>>lI,

5 awm’2&+l), flm’2&+l)>ll: 22 2Ca2(o - l)-‘6,’ exp(Mpk(a - a-2)6,)((u(t,), u(t,)>l;t

I C&,‘Npm m

with C = C(A4) independent of m. Then applying (21) with s = tm+l, t = t E [T - S,,, , T],

and p = ap, = pm+l we arrive at

5 exp(~p~+,~~,+,)l’Pm+ll(~(t,+~), Wm+l))lp,+l I (C&l)l’pmNm

I (CG~~)~‘~~N~

with a constant C = C(M) independent of m. This recurrence relation yields

N,,, 5 Pm&No,

where

m-1 m-l m-l

K??I = p~’ c o-j and A, = n (oj)2/o+J) = exp (log a2’Po) C ja-j

j=l j=l j=l >

.

1198 P. BILER et al.

Evidently, SUP,K~ + sup,&,, < 43, so the boundedness of I@(T), v(T)>], 5 sup,iV,+r I C(SUP[,,.]l(U, v>lpo) < 00 follows.

(iv) As it concerns the local in time continuity of solutions in rpO(sZ) the proof is analogous to that of their boundedness. Formally, we multiply the differences of equations (1) and (2) written for ur , u2 and ur , u2, respectively, with Iur - u~]~-~(u~ - u2) and I u1 - u2~p-2(u1 - u2), respectively. After one integration by parts we obtain the differential inequality

; Iul - u21; + 4(pp l) IV(lu, - u,1”“>1;

5 2(p - 1) (1

* 1% - ~21p’21vd% - ~21p’2m7d

+ s

l~,llW, - VzN% - ~21p’2-11w41 - u21p’2)l 9 0 >

and a similar inequality for ui - v2. Now, using the L”(Q) boundedness of Vp, and u2 just obtained, we follow the scheme of the proof of theorem 2 (iii).

Now, after establishing the L” smoothing effect typical for parabolic equations, further regularity of solutions is a consequence of the standard theory in, e.g. [13].

In particular, theorem 10.1 in Chapter III of [13] gives the Holder continuity: u, v E c a*01’2(Q x (0, T)) with some CY E (0, l), and the Holder constant is locally bounded in (x, t). Under assumptions on the supplementary regularity of the boundary aQ E C2+Ol, and the Holder continuity of (u, u) restricted to the parabolic boundary of Sz x (0, T) (with suitable compatibility conditions on a0 x (01) we can obtain C2+a*‘+a’2(G x [0, T]) regularity of solutions, which are in this case the classical ones, cf. e.g. [13, Chapter III, Section 121.

Remark 3. Some results in this section can be proved under weaker assumptions on the boundary asZ of the domain Q. For instance, local existence results are still valid for n I 3 under the standard assumption in [13] on a piecewise C’ boundary. Local regularity results in theorem 2 (iii) hold true for Lipschitz domains in lR”, n 5 3. The proof, however, should be slightly modified for small p; the crucial estimate is Iv~]~+~ < 00, with some E > 0, for solutions of the Poisson equation (3), (6) in Lipschitz domains. This is a recent result of David Jerison and Carlos Kenig. We are grateful to Carlos Kenig for the information on this fact (personal communication).

Note that theorem 2 (iii) applies to both cases: electrolytic and gravitational, so a local in time character of each LP(C2) regularity is the best possible result. Theorem 6 has some consequences for the global in time regularity in Lp(sZ), including p = 00, but it is proved under the assumption sup,, 0l(u(t), u(f))12 < co, which is automatically satisfied for n = 2 (see theorem 3), but it is not known whether this always holds for n = 3.

Observe that standard maximum principles are not available for the system (l)-(8); the boundary conditions (4), (5) do not provide any pointwise control of (u, u) on a0 x [0, T].

3. GLOBAL IN TIME EXISTENCE OF SOLUTIONS

The results in this section are new compared to those in [9], even for the simplified model considered therein.

Debye system 1199

THEOREM 3. If Sz is a bounded Cl+’ domain in R2 and u,, 1 0, u0 2 0, uO, u0 E &(a), then the unique solution (u, v) of problem (l)-(8) is global in time. Each component U, v of this solution belongs to the space L”((0, ~0); L’(Q)), and Vu, VU E &((O, 00); L2(n)).

Remark 4. This result generalizes and improves that in remark 8 in [9], where the global existence for one parabolic equation version of problem (l)-(8) has been proved for small initial data only.

The global existence result in theorem 3 depends heavily on the electrolytic character of equations (l), (2) (the signs -, + in the transport terms, respectively). In the gravitational two-dimensional radial case (the sign + in (l), Y = 0) it can be proved that for IuO(i > 8n solutions cannot exist globally in time. This phenomenon studied in [ 1 l] can be interpreted as a gravitational collapse.

Proof of theorem 3. The inequality obtained by summing (9) and (10) reads

5 tMlv42 + l~l3lv42mla

5 CIV(u, elzle4 v>lJu - ~l:‘21v,l:‘2.

For an estimate of the L3(sZ) norm we will need the following inequality valid for every E > 0, some C, = C(E, 0) and each w E H’(Q), a C lR2

Id: 5 h4lfl~ld4l, + Gl4,. (22)

Before proving this, we note that (22) allows us to estimate the “bad” term in the two- dimensional case when the net smoothness coefficients, mentioned in remark 2 above, for the “bad” and “good” terms coincide, by making the critical part of the “bad” term small enough. We will use the fact that the quantities /(u(t), v(t)>(l, (u(t) log u(t)/ 1, Iv(t) log v(t)], ,

IVp(t)12 are a priori bounded for all t 1 0 by the L2(sZ) norms of the initial conditions. This follows from the conservation of charges property, and from the analysis of the Liapunov function W(t) = jn (u log u + tr log u + tlV912) dx which decreases along the trajectories of (l)-(8): W(t) I W(O), see Section 5, formulas (32), (33) in this paper. It is worth noting that the gravitational system mentioned in remark 1 does not have such a good Liapunov function.

Consider a number N > 1 and the function x defined on R by X(S) = 0 for IsJ I N, t(s) = 2(14 - N) for NC IsI 5 Uv, and x(s) = IsI for IsI > 2N. Observe that

md

IX(WNI 5 (,w,~N) iwl s (bN)-’ s

(,w,~N) lwl lodwl 5 (lN’O-‘b~dw~l,.

1200 l’. LIILbK et al.

Moreover, the H’ norm of X(W) can be estimated by

Ilx(w)llf = IV~XWI~ + Ixwl; 5 IX’W w +

From the Sobolev inequality for n = 2 we obtain

Ix(w)l; s 4lW + lwlf 5 4llwllT.

hence,

Iwl; 5 C(10gN)-‘((w~~:.~w10gIw~(, + 4N21wl,.

Taking N = exp(C/a) and C, = 4 exp(2C/e) inequality (22) is proved. Using (22) we can continue the proof of theorem 3

; $104, u>1: + IV(w U>lf

5 cIvdl’211w, ~>ll1(~ll(~, u>Il:(lu log UII + lulog UII) + c,l<w v>11Y2

s tll<u, NT + M(u, dl,,

where E > 0 is chosen suitably small, according to the initial value of the Liapunov function W, in proposition 3, e.g. E = (8CW,(0))-2, and M = M(uO, u,,, E).

Finally, we obtain

-$l(u, u>li + 2l(u, u>l”z + lV(u, u>l;

5 3kw u>Ii + A4 5 a<& U>ll,I(% U>ll + M 5 ; Ilw, u>llT + Cl<% U>lt + hf.

This differential inequality leads to the uniform boundedness of I(u(t), u(t))12 since d/dt w + w + 19 5 A4 for w(t) = ((u(t), u(t)>l~, a(t) = $(V(u(t), u(t)>lz, and a constant M 2 0, which implies w(t) I M + w(O) and 1; G(s) ds < 00 for all t r 0.

The proof is finished by recalling the construction from theorem 1 which can be used tc obtain the continuation of the local in time solutions to the global ones with an L2 bounc uniform in time.

The next result concerning global existence in the three-dimensional case is a generalizatior of proposition 4 in [9]. We postpone its proof to the Appendix because properties of stationar: solutions described in the next section will be an important ingredient of the proof of thei asymptotic stability with respect to perturbations which conserve charges.

THEOREM 4. Let SJ c lR3 be a bounded Cl’& domain. There exists E > 0 such tha given (u,, u,) if (U, V, @) is the stationary solution of (l)-(6) (see also (28)-(30) below satisfying lull = IdI, 11/l, = luoll and lug - UI, + Iv0 - VI2 < E, then the solution of thl problem (l)-(8) exists for all t 2 0. Moreover, [(u(t), u(t)>j2 is uniformly bounded, am lim,,,l(u(t) - II, u(t) - V)(, = 0.

Debye system 1201

The last part of this section deals with radial solutions of (l)-(8). Now our problem is studied in Q = B-the unit ball in R”, n r 2. The main result is the following theorem. The second part of its proof will be postponed until the Appendix for the same reason as that for theorem 4.

THEOREM 5. Given p > n, u. 2 0, u,, z 0 which are radially symmetric functions from P(B), B c IR”, there exists a unique radial solution of (l)-(8) which is global in time: (u, v) E (L”((0, w); L!‘(B)))‘. Moreover, this solution converges in L”(B) as t 4 to to the radial stationary solution ((I, V> with IUI, = IuOll, /VI, = Iv,Ji.

Proof. In the radial case CI = B it is possible to obtain an a priori estimate for the norm (of the solution (u, u) constructed in theorem 2) [(u(t), u(t))lp uniform in time, and, hence, the continuation of solutions to the global ones is possible. First, note that the system (l)-(3) written as

ut = Au - Vu - Va, - u(u - u)

u, = Au + Vu - Vy, + u(u - u)

Aq=u-u

is evidently rotation invariant. Hence, for radially symmetric initial conditions the unique

solution constructed (in the class of weak solutions considered in theorem 1 or 2 (i)) is radial. As it concerns its boundedness, the formal argument goes as follows. Let us multiply (1) by up-‘, (2) by VP-‘, and integrate over Q = B. After two integrations by parts we get

g 14; + 4@; l) Jv(tP2)l; = p I * V(uP_‘) * (u Vv)

= (P- 1) I

V(uP) . vfp n

= (P- 1) up:-(p- 1) s

up Ay, (23) an n

and

g 14:: + 4’Pp- l) Iv(vp’2)l; = -p I V(T’) * (UVV)) 0

= -(p - 1) z?$+(p- 1) VP Ap. (24) an n

The crucial observation is that in the radial case a~/& on aQ is a constant depending on in (uO - uO) = ju Ap only. So, summing up (23) and (24) we arrive at

; ((u, u)lpp + 4(pp l) JV(lP2, up'2)l:

SC I (UP + u”) - (p - 1) (UP - vp)(u - u) ae 5 B

5 Cl ( Up'2, uP’2) liZ(aBj I C/l ( Up'2, lY’2) (I f/2, (25)

1202 P. HLhK et al.

with the last inequality following from the Sobolev trace theorem. Next, interpolating Sobolev norms, we obtain from (25)

Finally, the differential inequality

$ IO4 VA:: + ; 104, v>I$ 5 Cl<% 4,2

5 Cl(u, v)p-2)‘(p-1)I(u, v)Iy(p-l)

I ; I(u, u>l”, + Cl(u, v>lT

5 a I(u, v>I; + hf (26)

leads to the uniform in time boundedness of the Lp norm of (u, v> similarly as in the concluding part of the proof of theorem 3. Of course, this is sufficient to get the continuation of solutions to the global ones which stay in Lp(B).

4. STATIONARY SOLUTIONS OF THE DEBYE SYSTEM

In this section we recall some results from [lo] presented in a manner suitable to interpret the Poisson-Boltzmann equation as the stationary version of the Debye system. We begin with the observation that the stationary equations (l)-(3) for the quantities CJ, V, @

AU - v * (UVQ) = 0, AI’+ V.(I’VQ) = 0, A@==-V (271

are equivalent (since in the class of solutions considered in theorems 1 and 2 U, V E H’(Cl), and the potential @ belongs to Hi(Q) tl L-(Q)) to

V - (exp(@) V(exp(-Q)U)) = 0 (28:

V - (exp(-a) V(exp(Q) V)) = 0 (29:

AQ,=U- V, (30

(and for classical solutions the boundary conditions (4)-(6) can be written as

Ww-@W) = o

8V ,

a(exp(@) V o

av =) Q, = 0).

Debye system 1203

Taking the inner product of (28) and (29) with exp(-@)U and exp(Q)v, respectively, we obtain

s exp(~(x))lV(exp(-~(x))~O)12 dw = 0, n

r exp(-Q(x))lV(exp(@(x))V(x))J2 dx = 0. D

This means U(x) = p+ exp(@(x)), V(x) = p_ exp(-a(x)) a.e. in x E M, for some constants p+, p_ 2 0. The electric potential Q E H&2) satisfies the Poisson-Boltzmann equation

A@ = p+ exp(Q) - p_ exp(-a) (31)

(and the homogeneous boundary condition (6) Q ban = 0 if @ is a classical solution).

PROPOSITION 2. If a is a bounded domain in fR”, n 2 2, then for every U,, V0 > 0 there exists exactly one weak stationary solution (U, V, (D) of (28)-(30) such that so U(x) dx = U,, jn V(x)& = &.

If a is a regular domain with asZ E Cl+&, then (U, V, @) is the classical solution.

Proof. For 17, 1 V, equation (31) can be written in the form

which is a particular case of the Poisson-Boltzmann equation studied in [IO]. When U, < V. , then it suffices to replace Q, by -@‘, U by V and V by U. Although [lo] has only dealt with the three-dimensional case, a generalization of reasonings in [lo] to the cases n = 2 and n 1 4 is straightforward. If Q has the C2 boundary, then the existence of classical solutions is a consequence of the Leray-Schauder theorem applied to a suitable integral operator.

The uniqueness of solutions follows from monotonicity arguments in [lo], still valid for weak solutions Q, E H,‘(a) fl L”(n).

Note that the C2 regularity of the boundary asZ of the considered domain in [lo] has been used to get uniform estimates of the gradients of (continuous up to iX2) solutions of the Poisson equation with a continuous right-hand side, and this assumption can be relaxed to Cl+’ regularity, cf. e.g. [15, Chapter II, Section 4, corollary 81.

The general case is proved by exhausting the domain fi by an increasing sequence of smooth domains Qk for which (31) with the condition @k = 0 on a& is solved. Then the estimates for solutions ak from [lo]

[V@& I C& and l@kL 5 CL,

nold, where lk is the first eigenvalue of the Dirichlet Laplacian -AD on &, and C is a constant ndependent of k. Since & I A, for all k, the sequence (@,J contains a subsequence convergent weakly in H,‘(Q) to an element ‘.I) of H:(a) fl L”(Q). It is standard to verify that Q is a weak ;olution of (31).

1204

5. LARGE TIME BEHAVIOR OF SOLUTIONS-CONVERGENCE TO STEADY STATES

We begin this section with the verification that the system (l)-(S) has a Liapunov function. The existence of such a Liapunov function is not affected by our nonlinear boundary conditions (4), (5) and this function resembles those in [5, 7, 91.

PROPOSITION 3. The function

W,(t) = s (u(x, i)(log u(x, t) - 1) + 1 + v(x, t)(log u(x, t) - 1) + 1 + #Q!?(x, tp) dx n

is a Liapunov function in the weak sense for the system (l)-(8), i.e.

$ w,(l) + s

04x, thwx3 w, 0 - 9(x, 0)12 + v(x, mwx w, 0 + 9(x, 0)12) dx 5 0 D

(32)

holds for every weak solution of (l)-(8), W,(t) 2 0, and d/dt W, < 0 outside stationary solutions.

Remember that Liapunov functions in the strong sense satisfy the differential inequality d/dt W + EW zs 0 with some E > 0.

Proof. It repeats the arguments in the proof of lemma 3 in [9]. The function W, is approximated by the functions W, ,6 + O+, defined similarly as W, but using (U + 6), (v + s) instead of U, o. Such a regularization enables us to treat log u, log v harmlessly (after replacing them by log@ + 6), log(u + S) and then passing to the limit 6 -+ O+).

We note an important consequence of (32) which has the character of an a priori estimate

;wpt) 1% owl, + lw log WI, + IV9WI2) < c0 (33)

for uO, u, e L log L(Q), so in particular for u ,,, u. E L2(sZ). Observe that the supremum in (33) only depends on u. , u. .

For globally defined (for all t 1 0) solutions of (l)-(8) which are uniformly bounded in L’(Q standard arguments (cf. [5,9]) show the following theorem.

THEOREM 6. Assume that Q C IT?, n = 2, 3, an E Cl+‘, and (u, v) is a weak solution of tht problem (l)-(8) with the initial conditions 0 5 uo, u. E L2(sZ).

If sup,,,j(u(t), u(t))l, < 00 (this assumption is always satisfied when n = 2, see theorem 3) then

limI(u(t) - U, v(t) - V)I, = 0, t+m

where (U, V) is the (unique) stationary solution with the same charges as the initial condition for(~,r.+:]nU(x)dx=jnuo(x)dx,jnV(x)dx=]n~O(~)dx.

Proof. First we shall show

lim(l(u(t) - II, v(t) - V>li + ll9W - @III) = 0. t-+m

(34

Debye system 1205

The proof is patterned upon that of theorem 2 in [9]. After a slight modification of the Liapunov function W, in proposition 3 we arrive at

$ w(t) + s

(MX, M(log u(x, t) - 9(X, 0)12 + &G 0lV(log u(x, r) + 9(x, 0)12) dx 5 0 (35) R

where

w(t) = .i

(4x, t)(hwx, t)/wx)) - 1) + W) + 4% t)(lwz(v(x, t)/w)) - 1) n

+ V(x) + 3lW(x, t) - W))12) k.

After the integration of (35) over [0, T], and then letting T tend to 00, we conclude that there exists a sequence tk + 00 such that

3 04% tk) I V(log 4x3 fk) - 9(x, tk)) I 2 + w, tk) I vx 4& a + m &)I I 2, du n

= I dv(2u1’2(x, fk)) - u1’2(x, t/J VP@, fd12 B

+ IV(2v1’2(x, tJ) + ?P2(x, fk) V&x, t/J2) dx (36)

tendstoOask+oo. The Liapunov function W*(t) 2 0 from proposition 3 is bounded, hence, a subsequence

V9(tk), with a new subsequence t, + 00, still denoted tk, converges weakly in L2(sZ). Taking into account the boundary condition (6) (9 = 0 on 8Q) we see that v)&) converges strongly in L2(Q) to a function v/.

We claim that

S~PlVU”2&)129 sup(vu”2(tk)(2 < 00. (37) k

If (37) is satisfied, then from the compactness of imbeddings H’(Q) CC L*(Q), 3 c Q < 6, N’(a) CC Lr(X’l), 2 < r < 4, we conclude that there exist functions w, z E Lq(CJ) rl E(&2) such that w = limk,, u”2(tk), z = limk,, U”’ (tk) in the above space along a subsequence of t,s. In particular, we get w2 = limk,, u(tk), z2 = limk,, u(tk) in L’(a).

From the weak convergence of gradients in (36), and from (37) we infer 2 VW = w Vy/, 2 Vz = --z Vy/, 2 V(log w) = Vy/, 2 V(log z) = -VW. This permits us to identify (w2, z2, I& with a stationary solution (U, V, @), using the result in proposition 2.

Now, limk,, W(tk) = 0 and nonincreasing of W along the trajectories show lim,,, W(t) = 0, and in particular lim,,,Il9(t) - @‘(Ii = 0 f o 11 ows. The L’ convergence of u(t) and v(t) is implied by the estimate

lu(t) - uI1 = 5 lu(x, t) - U(x)1 dx I n

In@+ co~~loswu~d.s)dx 5 sln( + C(6)

1 (u(log(u/U) - 1) + U)dX

0

1206 P. BILER et al.

where 6 > 0 is taken sufficiently small. The second term above is controlled by the Liapunov function w(t) -+ 0. An analogous reasoning shows that lim,,,]v(t) - VI, = 0, too.

We return to the proof of the claim (37). Clearly, from (36) it suffices to show that

with M independent assumption on (u(t),

2 j

VU. Vv 5 + n

o ]Vu(‘u-’ + M,

of t (and similarly for u). This is a consequence of the boundedness u(t)> since

j VU. VP I (j IvU12U-r~‘2(j UlVpl)l/2

,a51VU12U-‘+ jUlVP12

5 a ]vu(%-’ + lul,(v& j

5 i s

IVU]~U-’ + M(u121u - 01;‘”

with M = M(sup]Vy,],). This concludes the proof of the claim (37), and therefore (34) is now proved.

Remarks 6 and 7 from [9] on various types of convergence to the stationary states under weakened/strengthened hypotheses on (u, u> apply to problem (l)-(8) mutatis mutandis.

Next we improve the result (34). As a by-product of the proof we also obtain some global in time regularity of weak solutions. These results are meaningful and new even for the simplified model in [9].

First, using the L2(Q) uniform in time boundedness of solutions, we obtain from theoren 2 (iii) the following property: U, ZJ E L”(Q x (6,~)) for every 6 > 0.

Then we can write

KU - U, u - v>l:: 5 a04 u>L + I(& ~)I.x’l(~ - u, u - ol,,

which implies together with (34) the convergence to the stationary states in the P(n) norm fo: every finite p.

The conclusion on the uniform in x convergence follows from theorem 2 (iv) sine (u(t) - U, u(t) - 1/) is controlled in the L”(Q) norm by its P(D) norm for p > n.

Remark 5 (cf. remark 9 in [9]). Similar existence results can be proved with essentially no nev ideas involved, for more general systems, where the, so-called, recombination terms R(u, u) o at most linear growth in u and u, cf. [5, 61, are introduced into equations (1) and (2). Of course such terms would modify the structure of stationary solutions and their attractivity properties Moreover, generalized Debye systems with several different kinds of ions can be treated in completely analogous way.

Debye system 1207

Condition (6) for the potential v, can be replaced by another physically reasonable linear

boundary condition, e.g. that of electrically isolated walls of the container a. In such a

situation all the reasonings can be modified inessentially, for example by replacing the Green

function in the representation of ~1 by the fundamental solution of A or another related kernel.

Acknowledgements-The authors are much indebted to Danielle Hilhorst and Andrzej Krzywicki for fruitful discussions and constant interest in related problems. We are also grateful to Herbert Amann and Jan Goncerzewicz for their remarks.

1. 2.

3.

4.

5.

6.

7.

8.

9.

10. 11.

12. 13.

14. 15.

16.

REFERENCES

DEBYE P. & HiiCKEL E., Zur Theorie der Electrolyte. II, Whys. Zft. 24, 305-325 (1923). FRIEDMAN A. & TINTAREV K., Boundary asymptotics for solutions of the Poisson-Boltzmann equation, J. d@ Eqns 69, 15-38 (1987). KRZYWICKI A. & NADZIEJA T., A nonstationary problem in the theory of electrolytes, Q. A@. Math. 50, 105-107 (1992). GAJEWSKI H., On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors, Z. angew. Math. Mech. 65, 101-108 (1985). GAJEWSKI H. & GROCER K., On the basic equations for carrier transport in semiconductors, J. math. Analysis Appfic. 113, 12-35 (1986). MOCK M. S., An initial value problem from semiconductor device theory, SIAM J. Math. Analysis 5, 597-612 (1974). MOCK M. S., Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Analysis Applic. 49, 215-225 (1975). SEIDMAN T. I., Time dependent solutions of a nonlinear system arising in semiconductor theory-II. Boundedness and periodicity, Nonlinear Analysis 10, 491-502 (1986). BILER P., Existence and asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux boundary conditions, Nonlinear Analysis 19, 1121-l 136 (1992). KRZYWICKI A. & NADZIEJA T., Poisson-Boltzmann equation in @, Ann. Pal. math. 54, 125-134 (1991). BILER P., HILHORST D. & NADZIEJA T., Existence and nonexistence of global solutions for a model of gravitational interaction of particles, Colloquium Math. (to appear). ADAMS R. A., Sobolev Spaces. Academic Press, New York (1975). LADYZENSKAJA 0. A., SOLONNIKOV V. A. & URAL‘CEVA N. N., Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, Rhode Island (1988). LIONS J.-L., Quelques mkthodes de rtkolution des probkmes aux limites non linkaires. Dunod, Paris (1968). DAUTRAY R. & LIONS J.-L., Mathematical Analysis and Numerical Methods-for Science and Technology, Vol. 1. Springer, Berlin (1990). RUBINSTEIN I., Counterion condensation as an exact limiting property of solution of the Poisson-Boltzmann equation, SIAM J. Appl. Math. 46, 1024-1038 (1986).

APPENDIX

In this appendix we return to the proofs of theorems 4 and 5 on the global existence of solutions with the initial data

:lose to stationary states, and on the asymptotic behavior of solutions in the n-dimensional radial case. Moreover,

.emarks on the Debye system in an annulus will be given.

Proof of theorem 4. Proceeding analogously as in [9] we obtain the following a priori estimate for (u - (I, u - V)

2 &I@ - (I, u - v>1: + IV@ - (I, u - 01;

5 CIV(u - u, v - V)l,l(u - u, v - V)J,JV(co - @)I, + ClV(u - u, v - V>l&4 - u, v - V)I,lV@I,

+ CIV(u - u, v - Y>I,I(U, V)l,IV(u, - @)I4

5 CJV(u - u, v - v)(;“J(u - u, ” - Y)I,JV(@ - @)I:” + CIV(u - u, v - v~1;‘41v@I,lv(y, - @,I:”

+ CIV(u - u, v - v>J:“(cu, V)(&B - @)I;”

I ; IV(u - u, v - v>t; + C/V@ - @)l$(u - u, u - v>l; + ClV(u, - @9l:dw: + ku, w:,. (Al)

1208 P. BILEK el al.

Now observe that IV@ - @)I2 is controlled uniformly in t by the quantity I@, - (I, u0 - I’)\, only. Moreover, for the stationary solution lV014 + j(U, V)], is finite, by the inspection of the proof of proposition 2. Thus, (Al) implies the

differential inequality

i I& - U, v - v>lZ + IV@ - u, v - v>1; 5 M,l(u - u, v - V>l! + M, (A2)

with some constants M, and M, decreasing to zero when the quantity ((u, - U, I+, - &‘)I, tends to zero. From the

PoincarC inequality applied to (u - U, v - V), which satisfies jn (u - U) = jn (u,, - U) = 0, jn (u - V) = ln (v, - V) = 0, we can write I(u - U, v - V)\, 5 CIV(u - U, v - V)I,. Consequently, (A2) leads to

dw dt+6w~MM,~4+Mz

for the function w(t) = ((u(t) - U, v(t) - V)l:, with some 6 > 0. Now, if M, and w(0) are small enough, i.e. when

I(u, - U, v,, - V)I, is sufficiently small, w(t) must be bounded for all t 2 0. In fact, from (A3) we infer

dw/dt (M, w4 - 6w + k&-1 5 1, and the polynomial P(w) = h4, w4 - 6w + M, has a strictly positive root w,, for

M, > 0 sufficiently small. The integral 12 l/P(w) dw diverges, so if w(0) < w,, then w(t) cannot blow up in a finite

time, and lim, _ m w(t) 5 w,, . This makes possible the continuation of the solution (u, u) for all t s 0 by theorem 1, tc a solution with sup,,,l(u(t), v(t)>], < m. Theorem 6 applies, and as the conclusion we have in particular lim,,,l(u(t) - (I, v(t) - V)I, = 0. This means that the stationary solutions of the dynamical system generated bl

(l)-(8) in (L’(Q))‘, restricted to constant charges subspaces ((u, v): lull = U,,, JvI, = V,), are locally asymptoticall)

stable. In fact, by theorem 6, the convergence is much better.

Proof of the secondpart of theorem 5. The conclusion on the convergence of solutions in the L”(B) norm in theoren

5 (the radial case) is obtained in a completely analogous way using the Liapunov function Wand adapting with mine!

modifications the proof of theorem 6. Instead of giving the details, we note that our study of radial solutions o

problem (l)-(8) has been based on the original equations (l)-(6). However, it is easy to write radial versions of thesl

equations in new dependent variables

! s

, U(r, t) = u= p “-‘u(p, 0 dp,

W) 0

V(r, t) = s s

r v= p “-%P> 0 Q,

W) 0

$~(r, t) = q(x, t), 1x1 = r. ‘u and V denote (up to multiplicative constants) total charges of ions contained in the ball o

current radius r. In these variables we obtain

U, = ‘u, - (n - l)r-I%, - r’-“(?I - V)U, (A4

VW, = 9, - (n - l)r-‘9, + r’-“(‘ll - V)V, (A!

since rn-‘+, = Cu - V. Together with suitably written boundary conditions (4)-(6) this system considered fc

sufficiently regular (‘U, 0) is equivalent to (l)-(8) (remember that radial solutions of (l)-(8) are unique!), but the direr

analysis of this system is by no means trivial because of singular coefficients r-‘, r’-“. Theorem 5 implies th;

Q(t), V(1) tend (uniformly in r) as t goes to CQ to the corresponding integrated densities of radial stationary solution

Remark Al. A physically relevant problem (the evolution case of [2, 161) appears when equations (l), (2) are considere

in a domain Q of the form 0 < R, < r < R, CC m, i.e. in an annulus for n = 2, or in a spherical shell for n 1 3. In sue a situation the natural boundary conditions for the potential (instead of (6)) are

&(R,, t) + OR;-” = 0, HR,, t) = 0, (A(

where D = U,, - V, > 0 is the difference of the charges. Now, the integrated densities ‘u, V satisfy the syste

analogous to (A4), (AS)

cu, = ‘1L, - (n - l)r_‘?I, - r’-“(2l - V)U, + CT+W,

9, = VW, - (n - l)r-‘V, + r’-“(Tl - V)V, - ur’-“V,

Debye system 1209

with the boundary conditions

?.t(R,, t) = 17(&, t) = 0, ‘u(R,, t) = Us = const., 17(Ri, t) = V, = const.

It is relatively easy to see that this system has classical solutions. The regularity of solutions follows because the coefficients are no longer singular as it was in the case of a ball, cf. [13, Chapter VII, Section 71. Further properties

of the Debye system in annuli include the global existence of solutions (ZL, %‘) and their convergence (guided by a

Liapunov function) to the (unique., 1 cf. 121) stationary states. In fact, in [2] stationary solutions of (l)-(5), (A6) have

only been considered in the two-dimensional case. Here we sketch a proof of the existence of the radially symmetric

stationary solution in the n-dimensional case, n B 3. We will use the Leray-Schauder theorem, while in [2] variational

methods have been used.

The stationary solutions U, V have the Boltzmann form U = p+ exp(Q), V = p_ exp(-CD) (cf. Section 4). with the

potential @ satisfying

(Y’@‘(r)) = r”_‘f(@)(r) (A7)

@‘(R,) + OR;-” = 0, @(RX) = 0, (A8)

where

f(Q) = u” pexp(0) - v,

5 e exp(@) In exp(-@) exp(-a), u, > v,.

Note that the denominators in the expression f(@) are constants, so @ satisfies an ordinary differential equation with

a strictly increasing right-hand sidef(@). Integrating (A7) over [R,, R,] we obtain @‘(R,) = 0. Problem (A7), (A8) in

a new variable s = r2-n reads

@W(s) = (n - 2)-2,(2n-*,‘(2-n?f(~)(s) (A9)

f#J’(S,) = a/(n - 2), (AlO)

4&S) = +‘(.%l) = 0, (Al 1)

where #(s) = a(r), S, = R$-“, So = R:-“. We claim that f($(.r)) > 0 and b’(s) > 0 for all s E (S,, , S,]. Evidentlyf(0) # 0, otherwise the unique solution 4 = 0

of (A9) and (Al 1) does not satisfy (AlO). Iff(0) < 0, then we would have +“(S,,) < 0. Together with (AlO) this implies

b’(s) < 0, +(s) < 0, f(+(s)) < 0 on an interval (S,, FJ with S, < S < S, , &I’(.?) = 0. But this last relation contradicts the

equality

e5’(@ = (n - 2)-z J’r

r(2”-z)‘(z-n~(+(r)) dr. so

Therefore, f(0) > 0, and a similar reasoning proves our claim. Returning to (A7) we see that Q” = -(n - l)r-I@’ + f(@) > 0, so -aRim” 5 0’ I 0 and, thus,

0 5 @ 5 aR;-“(R, - R,).

Integrating (A7) twice over [r, R,] we obtain its integral form

RI @(I) = s”-‘f@(s)) d.sdr =: Z(Q)(r).

J

6412)

The operator 3 is a continuous compact operator in C[R,, R,]. The a priori bound (A12) allows us to apply the

Leray-Schauder theorem, so there exists a solution of the equation S(0) = a. The uniqueness of solutions of (A7),

(A8) can be proved exactly as in [2]. Note that the gravitational radial case in balls and in annuli is studied in [l 11.