Arch. Rational Mech. Anal. 116 (1991) 91- 99. �9 Springer-Verlag 1991

Asymptotics in Semilinear Periodic Diffusion Equations with Dirichlet

or Robin Boundary Conditions

P~TER HESS

Communica ted by H. W~nqB~e.CER

1. S t a t e m e n t o f t h e R e s u l t

In this note we study the behaviour as t ~ co of solutions of periodic boundary value problems of the form

3ru - k ( t ) &u = re(x, t) h ( u ) in O x (0, oo), (,) Bu = 0 on 0 O x (0, oo).

We assume that f2 C rR N (N _>_ 1) is a bounded domain with boundary 0f2 of class C 2+~, that k is a positive HOlder-continuous and T-periodic function ( T > 0 a given period), and that m is HOlder continuous in (x, t) e s~ x tR and T-periodic in t. The function h is assumed to satisfy

(H.I) h:fR 3 I--,R is concave and of class C 2.

We emphasize that h need not be strictly concave here. The following two cases are considered.

(H.2) I = [0, al and h(0) = h ( a ) = 0, h ' (0) > 0;

(H.3) I = [ 0 , oo) and h(O) = 0 , h ( s ) > 0 for all s > 0 ,

h(s) - - ~ 0 as s - ~ .

s

The nonlinearity m(x , t ) h ( s ) is thus partly concave, partly convex in s if m changes sign. The boundary operator B is either of the form

Bu = u (Dirichlet boundary conditions) or Ou

Bu = b ( x ) u + - - (Robin boundary conditions), On

where b E C 1+~ satisfies b ____ 0, b ~g0 on 3f2, and n denotes the exterior nor- mal to OQ.

92 P. HEss

By the parabolic maximum principle, the solution u of (*) is globally defin- ed with values in I, provided the initial values (at t = 0) lie in L We only con- sider such solutions. Note that we have classical solutions if the initial values are positive in s x (0, oo) (except for the trivial solution u ~ 0). We prove that each solution of (,) converges to a T-periodic solution as t -+ co, and that there is a globally asymptotically stable T-periodic solution of (*) except in a special case. The precise result is as follows.

Theorem. Each solution of (*) converges to a T-periodic solution, in C2(~) , as t ~ oo. (i) Let u = 0 be linearly stable. Then the trivial T-periodic solution u* = 0 is globally asymptotically stable (with respect to initial conditions having values in I). (ii) Let u = 0 be linearly unstable. Then there exists a (unique) positive T-periodic solution u* and u* is globally asymptotically stable (with respect to nontrivial in- itial conditions with values in l) . (iii) Let u = 0 be linearly neutrally stable. I f h(s) is not linear in some interval [0, s*] with s* > O, then u* = 0 is globally asymptotically stable. I f however, h(s) is linear on such an interval [0, s*], there exists a nontrivial one-parameter family {8~0 : 0 < e < s*] of T-periodic solutions, and each solution of (*) converges to an element of this family.

(The notions of stability employed will be recalled at the beginning of section 2.) I f 0f2 consists of more than one component, the boundary conditions may

also be of mixed type: Dirichlet, Robin and Neumann boundary conditions on individual components. We only exclude pure Neumann boundary condi- tions in the theorem. For nonlinearities satisfying (H.2), problem (*) under Neumann conditions has been studied extensively in [9], even in slightly more general form (Fisher's equation arising in population genetics). The arguments of [9] are however not applicable, e.g., in the case of Dirichlet boundary condi- tions. (Indeed, a crucial role in [9] is played by its equation (2.3), whose coef- ficient functions become unbounded near 0g2 for Dirichlet boundary condi- tions. Thus the boundary maximum principle fails in this case.) We present here an approach which is more in the spirit of [4], showing directly that each positive T-periodic solution is stable. Our approach also allows a simple and straightforward treatment of the Neumann problem provided h is strictly con- cave. If h is affine on some nontrivial interval, our techniques do not suffice in the Neumann case, and we need further arguments as applied in [9]. (The techniques of [9] permit m to depend nonincreasingly on u.) The present results are thus, in some sense, complementary to those in [9].

We finally remark that the question of asymptotics for periodic semilinear diffusion equations seems to be a rather difficult one. No Lyapunov function approach is available in general, and one has to deal with discrete-time order- preserving dynamical systems in the underlying functional-analytic setting. For such systems convergence results, which are not of a generic nature, are known only under stability hypotheses (e.g., [2, 11, 7, 1, 8]), or in spatial dimension one ([6, 5]). Our proof of the theorem makes essential use of the abstract

Semilinear Periodic Diffusion Equations 93

qualitative results on discrete order-preserving semigroups of [7] (which are collected also in [8, Chapter I]). We only give the proof for pure Dirichlet or Robin boundary conditions; it is clear how the arguments have to be modified in case of mixed boundary conditions.

2. Stability of Positive Solutions

We first recall the concept of stability of the T-periodic solution u of (.) (cf., e.g., [8, Section III. 23]). We look at the linearization

Otw- k( t) Aw = re(x, t) h ' (u(x , t)) w in s (0, co), (2.1)

Bw = 0 on 012 x (0, co)

and take the period map Qu of this equation, given by Q~[w(., 0)] = w(-, T). Then Q, is the FrOchet derivate [e.g., in the fractional power space E = XZ,

X = LP(t-2) > N, 2 + 2p < fl <= 1 associated to the LP-realization of

( -&, B)] of the period map of (*). It follows from regularity theory and the maximum principle that Q, is a compact, strongly positive operator in the strongly ordered Banach space E. The Krein-Rutman theorem implies that the spectral radius r = spr(Qu) is positive and is the principal eigenvalue of Qu, with associated positive eigenfunction w 0. The solution w(x, t) of (2.1) with initial value wo then satisfies w(. , t + 1) = rw(. , t) for t _ 0. The periodic solution u of (*) is said to be linearly stable if r < 1, linearly neutrally stable if r = I, and linearly unstable if r > 1. The linearized stability (linearized in- stability) of u implies the local asymptotic stability (instability) of u for the nonlinear system.

The stability can also be formulated in a different way. By the theory of the principal eigenvalue of periodic-parabolic problems (cf., e.g., [8, Section II. 14]), there is a unique number (eigenvalue) ~ (fir for which the linear problem

O t ( p - k ( t ) ~ x ( o - m ( x , t ) h ' ( u ( x , t)) (0=/~(o in (2x[R,

B~0 = 0 on 0f2 x ~, (2.2)

(0 is T-periodic in t

admits a positive solution (eigenfunction) (o. The relation between the spectral 1

radius r of Q~ and # is given by /~ --- - - - log r. Our main result in this section is T

Proposition 2.1. Let u be a positive T-periodic solution of (*). Then u is linearly stable or linearly neutrally stable. For Dirichlet boundary conditions u is linearly neutrally stable only if

IV~ul 2 h"(u) =--0 on ~x[0, TI. (2.3)

For Robin boundary conditions u is linearly neutrally stable only if both (2.3)

94 P. HEss

and the following condition hold:

b[h'(u) u - h ( u ) ] ~ 0 on OI2x[0, T]. (2.4)

As is readily seen from the proof, Proposition 2.1 holds also for Neumann boundary conditions.

We need the foUowing lemma (due to LAzER [10]) on periodic-parabolic eigenvalue problems (here r~ is a real-valued HOlder continuous function which is T-periodic in t).

/_emma 2.2. Let It E ~ be the principal eigenvalue of

Ot(o - k( t ) •(o - th(x, t) (o = It(o in I2x R,

B(o = 0 on 0s • ~,

~o is T-periodic in t

(with eigenfunction ~o > 0 on s215 R). Then It is" also the principal eigenvalue of the formally adjoint periodic-parabolic eigenvalue problem

- O t q / - k ( t ) A g / - m ( x , t) g / = I t ~ in 1 2 x ~ ,

Bg/ = 0 on 0s x R,

~u is T-periodic in t

(with eigenfunction g / > 0 on D •

Proof of Proposition 2.1. Let u be a positive T-periodic solution of (*), i.e.,

O t u - k ( t ) A u - m ( x , O h ( u ) = 0 in g-2• (2.5)

and Bu = 0 on 0 f2x~ . By Lemma 2.2, (with th = mh'(u)) , there exists the positive principal eigenfunction g/ of the problem adjoint to (2.2), i.e.,

- O t g / - k ( t ) A ~ - m ( x , t ) h ' ( u ) ~u=Itg/ in ~2x~ , (2.6)

Bg /= 0 on 0f2• ~, and ~ is T-periodic in t. We multiply equation (2.5) by h'(u) ~ and equation (2.6) by h(u) , and subtract to get

O,(h(u) g/) + k( t ) [ - A u . h ' (u) q/ + A~u. h(u)] = - I t h ( u ) ~u.

By the periodicity of u and g/ and by Green's formula, integration over QT = g2 • [0, T] gives

I I k(t)[<Vxu �9 Vx(h'(u) gt)) - <V,u/- Vxh(u)>] QT

Og2 ~n QT

Semilinear Periodic Diffusion Equations 95

The first integral in (2.7) equals

j j k(t) ~l Vxul2h' (u) �9 Or

The boundary integral in (2.7) vanishes in the case of Dirichlet (and Neumann) boundary conditions and equals

T

i i k(t) ~b[h'(u) u - h(u)] o of 2

in the case of Robin boundary conditions. Since I 1 h(u) q/> 0 and QT

h"(s) <=0, h'(s) s - h ( s ) < 0 for s~I , (2.8)

by the concavity of h, we infer that p >= 0. Moreover, for Dirichlet boundary conditions 12 = 0 only if

J VxulZh"(u) ~ 0 on (~T,

and for Robin boundary conditions only if this condition holds together with

b [ h ' ( u ) u - h ( u ) ] ~ O on 0f2 x [0, T].

(Indeed, q /> 0 everywhere on 3f2 • [0, T] for these boundary conditions.) []

Remark 2.3. Our assumptions (H.2) (H.3) each imply the existence of large positive periodic strict supersolutions a for (*). In case (H.2), the constant function t / = a suffices. In case (H.3), this follows from the sublinearity of h. Indeed, let Yl be the principal eigenvalue of the periodic eigenvalue problem

Otrl -- k(t) • r /= 7if/ in t2 x ~,

B~/= 0 on 0f2 x fR,

r/ is T-periodic in t.

Note that Yl > 0. Choose positive constants cl, c2 such that c 3 < Yl and

t)mllc((2T)h(s) <= ClS + c 2 for all s >__0.

Let a be the T-periodic solution of

a t a - k ( t ) A l l - - C l a = C 2 in f~x~,

B a = 0 on 00•

a is uniquely defined and positive by [8, Theorem 16.6], e.g., and is a periodic supersolution for (*). It can be made as large as desired by in- creasing c 2.

96 P. HEss

3. Proof of the Theorem

By Proposition 2.1, the positive T-periodic solution u of (*) is linearly stable except if

lVxul2h"(u) =-o

b[h'(u) u - h(u)] ~ 0

on QT,

on 0O • [0, T]

(for Robin boundary conditions), (3.1)

in which case u is linearly neutrally stable. This latter case can be characterized as follows.

Proposition 3.1. If the positive T-periodic solution u of (*) is linearly neutrally stable, then h is linear on some interval [0, s*] with s*>_ maxaT u, and the trivial periodic solution 0 is linearly neutrally stable. Conversely, if 0 is linearly neutrally stable and h is linear on [0, s*], with s* > 0 maximal, there exists a one-parameter family [e~ : 0 < t < s*} of positive T-periodic solutions of (*), and there is no other positive T-periodic solution of (*).

Proof. (i) Let u be a linearly neutrally stable positive T-periodic solution of (*). I f u satisfies Dirichlet boundary conditions, then IVxu I ~ 0 near 0g2x[0, T] by the parabolic maximum principle. We infer that h"(s) = 0 in some (maxi- mal) interval s ~ [0, s*], s * > 0. We show that s*_-_ max0_ u. Assume, to the �9

contrary, that s* < maXor u. Then there exists s such that s = u(x 0, to) > s* h"(s) < 0 for suitable (x0, to) E Qr. By (3.1),

I Vxu(x, t0)l = 0 for x in a neighbourhood of x0.

A continuity argument shows that u(x, to) = s for all x ~ ~ , contradicting the Dirichlet boundary conditions.

If u satisfies the Robin boundary conditions, then u ( x , t ) > 0 for all (x, t) ~ 0T. Since b (x~) > 0 for some x t E 092 and

b(xl) [h ' (u (xb t ) ) U(Xl, t) - h ( u ( x l , t))] = 0 for all t, (3.2)

it follows by (2.8) that h(s) is linear in s up to (maximal) s* >__ maxtu(xl , t). That s* ___> max0r u follows as above. Indeed, the contrary assumption would imply that u(x, to) = s > s* for some to and for all x ~ ~ . Taking x = xl, by (3.2) we would conclude the linearity of h up to this s, contradicting the maxi- reality of s*.

Hence in either case, h ( s ) = h ' (0)s for sE [0, s*], and since u > 0 now satisfies the linear (eigenvalue) problem

Otu -- k( t ) Au - re(x, t) h'(O)u = 0 i n O • ~,

B u = O on Og2•

u is T-periodic in t,

the linear neutral stability of the trivial solution 0 follows.

Semilinear Periodic Diffusion Equations 97

(ii) Assume, conversely, that 0 is linearly neutrally stable and h is linear on the maximal interval [0, s*], s* > 0. It is clear that each member of the one-parameter family {er : 0 < e =< s ' l , where ~0 is the (positive) principal ei- genfunction of the equation (*) linearized at u = 0 and normalized by max0r ~o = 1, is a positive T-periodic solution of (*). Since the principal eigenvalue p = 0 has multiplicity one, there is no positive T-periodic solution of (.) with values in [0, s*] other than those of the one-parameter family {e~0}.

We show that there is no other positive T-periodic solution of (*) at all. Indeed, assume for contradiction that u ~s*~0 is a T-periodic solution. By Remark 2.3 we find a T-periodic strict supersolution a with a > s*r a > u on Qr. By a standard procedure we arrive at the maximal T-periodic solution v: a > v >_ u. Since u 4: s*~0, v admits values larger than s* and is thus linearly stable by Proposition 2.1 and the above continuity argument. We embed (,) in the one-parameter family

3tw - k ( t ) ~ w = )~m(x, t) h ( w )

B w = 0

w is T-periodic in t

in s

on OOxN, (*~)

of periodic problems, with ~ near 1. Problem (.~) can be formulated as an abstract operator equation between the Banach spaces

6C + = ' 1 + 2 ( O x ~ ) : w is Tperiodic in t, B w = O on 0 D x ~ ) F I : = [w 2

and

F : = [w~ C ~'g ( ~ x ~ ) :w is T-periodic in t l ,

cf. [8, Section III. 26]. Since v is linearly stable, the implicit function theorem applies; for 2 < 1 close to 1 we obtain a positive T-periodic solution vz of (*~) which is still (linearly) stable. Furthermore, the trivial solution 0, which is linearly neutrally stable for ( . ) = (*0, becomes (linearly) stable for (*D, 2 < 1 (of. [8, Proposition 26.5]). By [8, Theorem 6.1] there exists an uns tab le T-periodic solution wz of (,z) between the stable T-periodic solutions 0 and vz (note that, e.g., 0 is an isolated T-periodic solution for (,~)). We assert that maxt~rw ~ > s*. Indeed, w~ _< s* would imply that w~ > 0 satisfies the linear problem which is the linearization of (,z) at w = 0. And this, in turn, would imply the linear neutral stability of the 0-solution of (,a), a con- tradiction. Thus m a x ~ r w ~ > s*, and we conclude again that w~ is a stable T-periodic solution of (*;~). With this contradiction, it follows that (,) admits only the T-periodic solutions etp, 0 <_ e <<_ s*.

This proves Proposition 3.1.

We are now ready for the

Proof of the Theorem. (i) Let the trivial T-periodic solution 0 be linearly stable. Any positive /Zperiodic solution v of (,) is stable by Proposition 3.1,

98 P. HEss

and 0 is isolated periodic solution. Theorem 6.1 of [8] implies the existence of an unstable positive T-periodic solution between 0 and v, which is impossible. Thus there is no positive T-periodic solution in this case.

That 0 is globally attractive follows by solving the initial value problem (,) with the initial data given by a large periodic supersolution: The solution u restricted to t = nT (n E ~q) is then decreasing and converges to 0 (as the only initial condition for a periodic solution); cf. Remark 2.3. The stability of 0 is then a consequence of the maximum principle (cf. also [8, Lemma 4.3]).

(ii) If 0 is linearly unstable, then any positive T-periodic solution of (,) is again stable by Proposition 3.1. The existence of a positive T-periodic solution is guaranteed by the presence of small positive T-periodic subsolutions and large T-periodic supersolutions; this together with the Theorem of [8] quoted above also implies its uniqueness and global asymptotic stability.

(iii) Let 0 be linearly neutrally stable. If there is no nontrivial interval [0, s*] in which h is linear, any positive T-periodic solution of (,) is linearly stable by Proposition 3.1. Assume that there is such a positive T-periodic solution v. We proceed as in the last part of the proof of Proposition 3.1 and embed (,) in the one-parameter family (,~). For 2 < I close to 1 there is a stable positive T-periodic solution v~ of (*~), and also 0 is stable for (*~). Again there exists an unstable Tperiodic solution of (*~) between these solutions, which is impossible by Proposition 3.1. Hence (*) admits only the trivial T- periodic solution. Its global asymptotic stability follows as in case (i).

If h is linear on some interval [0, s*] with s* > 0, we have the family of T-periodic solutions {efp : 0 < e _< s*], and no other T-periodic solutions, by Proposition 3.1. All solutions er are order-stable, and hence stable by [8, Lem- ma 4.3]. It is a consequence of the stabilization result [8, Theorem 3.3] that each solution of (*) converges to a Tperiodic solution, i.e., to an element e~, first in the norm of X/~ and thus (by standard estimates) in CZ(~).

All assertions of the theorem are proved.

4. Concluding Remarks

1. Proposition 2.1 also applies in case of Neumann boundary conditions

(B = fin) and gives an alternative proof of the following result on the periodic

Fisher's equation (cf. [9] and [8, Theorem 29.1]). Let h be strictly concave and satisfy (H.2), and exclude the case where

m = re(t) is independent o f x and ~ m ( t ) d t = O. Then there exists a (unique) T- periodic solution of (*) with values in I = [0, a], which is globally asymptotically stable (with respect to initial conditions Uo having values in I, uo~O, ~ a ) .

2. As a special case we obtain the corresponding results for the associated autonomous problem (convergence to stationary solutions), by letting the period T tend to 0. Our results then generalize those in [4], e.g., where strict concavity of h was assumed.

Semilinear Periodic Diffusion Equations 99

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Mathematisches Institut Universit~it Ztirich

R~imistrasse 74 CH-8001 Ziirich

(Received April 11, 1991)