Reaction-Diffusion Equations

Funkcialaj Ekvacioj, 30 (1987) 111-114

Decay Properties of Global Solutions of

Reaction-Diffusion Equations By Tomasz DLOTKO (Silesian University, Poland)

This note is a supplement to our previous paper [2]. Certain generalizations associated with [2] are given concerning the asymptotic properties of global solutions of divergence parabolic equations. The subject here is the quasilinear parabolic Dirichlet problem, a special case of that considered in [2, Theorem 1];

(1)

$u_{t}=¥sum_{i,j}¥frac{¥partial}{¥partial x_{i}}(a_{ij}(x)¥frac{¥partial u}{¥partial_{X_{j}}})+¥sum_{i}b_{i}(x)¥frac{¥partial u}{¥partial x_{i}}$

$+f(t, x, u, ¥nabla u)$

(2)

on

$u=0$

where $C^{1}(¥overline{¥Omega})$

$¥partial¥Omega$

,

,

$u(0, x)=u_{0}(x)$

,

, , -smooth bounded domain in symmetric and elliptic, and a continuous function $t¥geq 0$

$R^{n}$

$¥mathrm{x}¥mathrm{e}¥mathrm{O}$

$b_{i}¥in C^{0}(¥overline{¥Omega})$

(3)

$u¥cdot f(t, x, u, ¥nabla u)¥leq Cu^{2}$

, with satisfying;

$¥nabla u=(u_{x_{1}},¥cdots, u_{x_{n}})$

$f$

$ a_{ij}¥in$

.

The growth rate [2] for $(u ¥cdot f¥leq Cu^{2}+D)$ has been sharpened to (3). This is needed for the iteration procedure (as in Theorems 1, 2 of [2]) to give decay instead of only boundedness. Moreover is assumed to be Lipschitz with respect to (constant ), (constant ), differentiate with respect to ¥ const. (M); both these assumptions are , and bounded with $K_{M}:=R^{+} ¥ times ¥ overline{ ¥Omega}¥times[-M, M]¥times R^{n}$ . uniformly satisfied in sets In accordance with our convention [2] all sums are taken from 1 to and all integrals over . norms in (4) replaced by It is shown in [2] (with norms this is not essential for the proof in [2] that: $f$

$L^{¥infty}(¥Omega)$

$f$


$N_{1}$

$t$

$¥nabla u$

$N_{3}$

$u$

$|f| leq$

$¥partial f/¥partial u¥leq N_{2}$

$n$

$¥Omega$

$L^{1}$

$L^{2}$

$)$

Proposition 1.

(4) $t¥geq 0$

(5)

Conjunction $||u(t, ¥cdot)||_{L^{1}(¥Omega)}¥leq$

, ensures estimates

uniform

$||u(t, ¥cdot)||_{L^{¥infty}(¥Omega)}¥leq$

of the const.,

two conditions $||u_{t}(t, ¥cdot)||_{L^{1}(¥Omega)}¥leq$

const.

in time

const.,

$||u_{t}(t, ¥cdot)||_{L^{p}(¥Omega)}¥leq$

const.

?

Tomasz DLOTKO

112 $p¥geq 1$

, and finally (for some

(6)

$¥delta¥in(0,1)$

$||u||_{c^{1/2,12}}l(R^{+}¥times¥overline{¥Omega})¥leq$

const.,

We want to study here the We have: For some

Theorem A. $||u(t, ¥cdot)||_{L^{¥infty}(¥Omega)}¥rightarrow 0$

as

$ t¥rightarrow¥infty$

)


$q¥in N$

$||¥nabla u||_{C^{¥delta/2,¥delta}(R^{+}¥times¥overline{¥Omega})}¥leq$

const.

decay to zero of the solution

let

when

$||u(t, ¥cdot)||_{L^{q}(¥Omega)}¥rightarrow 0$

$u$

of $(1)-(2)$ .


.

Then

.

convergence together with First it must be remembered that the convergence for all $p¥in N$ (but not necessarily boundedness (5) guarantees convergence. with the same rate of decay). We proceed to the proof of the , which together is compact in As a consequence of (6) the family , decay to zero ensures the existence of a sequence of times with the . This gives uniform in decay in both in such that

Proof.

$L^{q}$

$L^{p}$

$L^{¥infty}$

$L^{¥infty}$

$¥{u(t, ¥cdot)¥}_{t¥geq 0}$

$C^{0}(¥overline{¥Omega})$

$L^{q}$

$ t_{n}¥rightarrow¥infty$

$C^{0}(¥overline{¥Omega})(L^{¥infty}(¥Omega))$

$u(t_{n}, ¥cdot)¥rightarrow 0$

$L^{r}(¥Omega)$

$r$

; $||u(t_{n}, ¥cdot)||_{L^{r}(¥Omega)}¥leq|¥Omega|^{1/r}||u(t_{n}¥cdot)||_{L^{¥infty}(¥Omega)}$

$¥leq¥max¥{1;|¥Omega|¥}||u(t_{n}, ¥cdot)||_{L^{¥infty}(¥Omega)}¥rightarrow 0$

.

, which is the consequence of our We need also the simplified version (with sharpened condition (3) of the estimate (number (8) in [2]) shown in Theorem 1 in this is a dynamical system we may change 0 into of [2], Since estimate, to get: $¥tilde{D}=0$

$)$

$t_{0}¥geq 0$

$u(t, ¥cdot)$

(7)

$y_{k}(t_{0})¥leq¥max¥{||¥mathrm{u}(t_{0}, ¥cdot)||_{L^{2^{k}}(¥Omega)}; y_{k-1}(t_{0})(¥tilde{B}+¥tilde{C}2^{(k+1)3n/2})^{2^{-¥mathrm{k}}}¥}$

where For arbitrary

$y_{k}(t_{0}):=¥sup_{t¥geq t_{0}}||u(t, ¥cdot)||_{L^{2^{k}}(¥Omega)}$

(i) (ii)

take

$¥epsilon>0$

$n=n_{0}$

,

.

, such that ,

$¥max¥{1, |¥Omega|¥}||u(t_{n¥mathrm{o}}, ¥cdot)||_{L^{¥infty}(¥Omega)}¥leq¥epsilon$

$||u(t, ¥cdot)||_{L^{1}(¥Omega)}¥leq¥epsilon$

for in (7) to the value and (i), (ii) that $t¥geq t_{n_{0}}$

Enlarging the constant follows from (7) with

$¥tilde{C}$

$(||u||_{L^{1}(¥Omega)}¥rightarrow 0)$

$¥tilde{C}’$

.

, such that

$¥tilde{C}’¥geq 2¥max¥{¥tilde{B},¥tilde{C}, 1¥}$

, it

$t_{0}=t_{n_{0}}$

(8) Then clearly

$y_{k}(t_{n¥mathrm{o}})¥leq¥max¥{¥epsilon, y_{k-1}(t_{n_{0}})(¥tilde{C}’ 2^{(k+1)3n/2})^{2^{-k}}¥}$

$y_{k}(t_{n_{0}})¥leq y_{k}^{¥prime}$

, where the increasing

$ y_{1}^{¥prime}=¥epsilon$

,

$y_{k}^{¥prime}=y_{¥acute{k}-1}$

$(¥tilde{C}’¥geq 1)$

.

sequence

$(¥tilde{C}’ 2^{(k+1)3n/2})^{2^{-k}}$

,

or by induction

(9)

$¥sup_{t¥geq t_{n¥mathrm{o}}}||u(t, ¥cdot)||_{L^{¥infty}(¥Omega)}¥leq¥lim_{k¥rightarrow¥infty}y_{k}^{¥prime}=:y_{¥infty}^{¥prime}$

$=¥epsilon(C^{¥prime}2^{3n/2})(¥prod_{k=1}^{¥infty}(2^{k})^{2^{-k}})^{3n/2}=¥epsilon C^{¥prime}(2P)^{3n/2}$

$=$

const. , $¥epsilon$

$¥{y_{k}^{f}¥}$

is given by;

Reaction-Diffusion

where

$x$

$P:=¥prod_{k=1}^{¥infty}(2^{k})^{2^{-k}}$

. Hence the proof is completed.

For a subclass of the equation (1) (with (exponent ) in ):

$a_{ij}¥in C^{1+a}(¥overline{¥Omega})$

,

$¥partial¥Omega¥in C^{2+a}$

,

$f$

Holder in

$K_{M}$

$¥alpha$

(10)

113

Equations

,

$u_{t}=¥sum_{i,j}¥frac{¥partial}{¥partial x_{i}}(a_{ij}(x)¥frac{¥partial u}{¥partial x_{j}})+f(x, u, ¥nabla u)=:Lu+f$

with the conditions (2) we have the following theorem concerning the decay of $||u_{t}(t, ¥cdot)||_{L^{¥infty}(¥Omega)}$

;

independent be the solution of (10)?(2) (with Theorem B. Let begining satisfying the conditions at the of this note) belonging to when . . Then $f$

$u$

of

$t$

$C^{1+¥gamma/2,2+¥gamma(R^{+}}$


$(||u_{t}(t, ¥cdot)||_{L^{1}(¥Omega)}¥rightarrow 0)¥Rightarrow(||u_{t}(t, ¥cdot)||_{L^{¥infty}(¥Omega)}¥rightarrow 0)$

$¥times¥overline{¥Omega})$

Proof. As a consequence of

the estimate (16) in [2] (with $N_{1}=0$ as a , $(c_{1}2^{k(1+n/2)}$ result of the special form (10) we have with . Enlarging to the value $c_{1}^{¥prime}=2¥max¥{c_{1}, c_{2},1¥}$ we arrive at the estimate 2 (for roots): $ z_{k}^{2^{¥mathrm{k}}}(t_{0})¥leq¥max$

$)$

$+c_{2})z_{k^{¥_}1}^{2^{k}}(t_{0})¥}$

$¥{¥int u_{k}^{2^{k}}(t_{0}, x)dx$

$z_{k}(t_{0}):=¥sup_{t¥geq t_{0}}||u_{t}(t, ¥cdot)||_{L^{2^{k}}(¥Omega)}$

$c_{1}$

$k$

(11)

$ z_{k}(t_{0})¥leq¥max$

$¥{||u_{t}(t_{0}, ¥cdot)||_{L^{2^{k}}(¥Omega)}, (c_{1}^{¥prime}2^{k(1+n/2)})^{2^{-k}}z_{k-1}(t_{0})¥}$

which is exactly similar to the estimate (8) for . Since we have also compactness of the family ( . estimate (22) in [2] or (12) below), then the same reasoning as in Theorem A finishes our proof. Remark 1. Sufficient conditions (related to the first eigenvalue of the , Dirichlet problem for the elliptic operator ) for which formulated in , were tend to zero when [2]. $u$

$¥{u_{t}(t, ¥cdot)¥}_{t¥geq 0}$

$¥mathrm{c}.¥mathrm{f}$

$L$

$||u(t, ¥cdot)||_{L^{2}(¥Omega)}$

$||u_{t}(t, ¥cdot)||_{L^{2}(¥Omega)}$


Remark 2. Often (cf. [1], [4]) solutions of parabolic problems tend to solutions of associated elliptic (stationary) problems. This situation holds for bounded solutions of (10)?(2) with independent of under additional . Compactness of trajectories ([2] Proposition 1) condition $L^{1}(¥Omega)$

$f$

$t$

$||¥mathrm{u}_{t}(t, ¥cdot)||_{L^{1}(¥Omega)}¥rightarrow 0$

(12)

$||u||_{C^{1+¥gamma l2,2+¥gamma}(R^{+}¥times¥overline{¥Omega})}¥leq$

const.

allows us to pass (on the sequence ) to the limit in the equation (10), to get consequence of Th. ): as a , , , ( ( ) $ t_{n}¥rightarrow¥infty$

$||u_{t}$

$t$

$¥cdot$

$||_{L^{¥infty}(¥Omega)}¥rightarrow 0$

$¥mathrm{B}$


$0=Lv+f(x, v)$ ,

where we denoted

$u(t_{n}, x)¥rightarrow v(x)$

in

$C^{2}(¥overline{¥Omega})$

$v=0$

on

$¥partial¥Omega$

,

.

There is also another way to prove the decay of the derivative solution. We have;

$u_{t}$

of the global

114

Tomasz DLOTKO

Proposition 2. Let be a global solution of the Dirichlet problem with continuous continuously differentiable in , , , and let . Then uniformly in when . $u$

$f$

$t$

$¥lim¥sup_{t¥rightarrow¥infty}¥partial f/¥partial u¥leq 0$

$u$

$¥nabla u$

$¥overline{¥Omega}$

$u_{t}¥rightarrow 0$

$(1)-(2)$

$¥lim_{¥mathrm{r}¥rightarrow¥infty}¥partial f/¥partial t=0$

,


Proof.

The proof follows from Theorem 1 p. 158 of [3]. It remains to note that since for global solutions the condition (12) holds, then composite , of the argument $(t, x, u, ¥nabla u)$ appearing in the equation , functions for are continuous and bounded uniformly in time. Then the proof is a direct consequence of the linear theory [3]. $¥partial f/¥partial t$

$¥partial.f/¥partial u$

$¥partial f/¥partial x_{i}$

$u_{tt}$

Below are listed recent publications close to our subject ([5], [6], [7]). References [1] [2]

Chafee, N., Asymptotic behaviour of a one-dimensional heat equation with Neumann boundary conditions, J. Differential Equations, 27 (1978), 266-273. Dlotko, T., Global solutions of reaction-diffusion equations, Funkcial. Ekvac., 30

(1987), 31-43. Friedman, A., Partial differential equations ofparabolic type, Prentice Hall, 1964. Hale, J. K., Stability and bifurcation in a parabolic equation, in Springer Lecture Notes 898, 1981, 143-153. [5] Lions, P. L., Structure of the set of steady-state solutions and asymptotic behaviour of semilinear heat equations, J. Differential Equations, 53 (1984), 362-386. [6] Lortz, D., Meyre-Spasche, R., and Stredulinsky, E. W., Asymptotic behavior of the solutions of certain parabolic equations, Comm. Pure Appl. Math., 37 (1984), 677-703. [7] Wiegner, M., On the asymptotic behaviour of solutions of nonlinear parabolic equations, Math. Z., 188 (1984), 3-22.

[3] [4]

nuna adreso: Institute of Mathematics Silesian University 40-007 Katowice Bankowa 14, Poland (Ricevita la 8-an de julio, 1985)