数理解析研究所講究録
913 巻 1995 年 1-16

1

Asymptotic Equivalence of A Reaction-Diffusion System to theCorresponding System of Ordinary Differential Equations(ある反応拡散方程式系と対応する常微分方程式系との漸近的同値性)
福岡大学理学部

星野(Hiroki Hoshino)

弘喜

1. Introduction

In this report Large-time behavior of a global solution to a reaction-diffusion systemwith homogeneous Neumann boundary conditions is studied. It is proved that the solution

behaves like the solution to the corresponding system of ordinary differential equations astime goes to infinity. This report is based on the paper [5] by Hoshino and Kawashima.
We consider the following initial-boundary value problem which stems from a model
for a simple irreversible chemical reaction:’,$t>0,$$x\in\Omega$,,$t>0,$$x\in\Omega$,$\frac{\partial u}{\partial t}=d_{1}\triangle u-uvmn$(1.1)$\backslash \frac{\partial v}{\partial t}=d_{2}\triangle v-uvmn$

,

(1.2)

(1.3)
Here

$t>0,$$\frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=0$$(u, v)(0, x)=(u_{0}, v\mathrm{o})(x)$$\Omega$

is a bounded domain in

outward normal derivative to$\mathrm{R}^{N}(N\geq 1)$$\partial\Omega$,$x\in\partial\Omega$$x\in\Omega$,.

with smooth boundary, the coefficients$d_{1},$$d_{2}$$\partial\Omega,$$\partial/\partial\nu$

and the exponents $m,$

denotes the$n$real constants satisfying(1.4)$d_{1},$and the initial data
(1.5)$(u_{0}, v_{0})(X)i$$d_{2}>0$,$m,$ $n\geq 1$

are bounded and nonnegative on$(u_{0}, v_{0)}\in L^{\infty}(\Omega)^{2}$

and $u_{0}(X),$$v\mathrm{o}(X)\geq 0$

forIn this report we may assume without loss of generality that
(1.6)$\overline{u}_{0}=\frac{1}{|\Omega|}\int_{\Omega}u_{0}(X)dx$,$\Omega$, that is,$x\in\Omega$.

$\overline{u}_{0}\geq\overline{v}_{0}>0$$\overline{v}_{0}=\frac{1}{|\Omega|}\int_{\Omega}v_{0}(x)dx$.

, whereare fixed2

If (1.1) is replaced with$\frac{\partial\tilde{u}}{\partial t}=d_{1}\triangle\tilde{u}-Ic1\tilde{u}^{m}\tilde{v}^{n}$,$t>0,$$x\in\Omega$$\frac{\partial\tilde{v}}{\partial t}=d_{2}\Delta\tilde{v}-Ic2\tilde{u}^{m}\tilde{v}^{n}$,$t>0,$$x\in\Omega$

,

$\{$

$m+nn/(m+n-1)$ and $k_{2}=I4_{1}^{\prime$I/(m+n\mathrm{s}_{2^{-}}r(m-1)/(m+n-1)$m}-1$, thenand we put $k_{1}=I\zeta_{1}^{-(1)/}-1$$K_{2}n-$satisfies (1.1); so that we deal with (1.1). Note that we have())$(u, v)=(k_{1}\tilde{u}, k_{2}\tilde{v})$$m/n$ in$I\mathrm{f}_{1}/I\mathrm{f}_{2}=$the case where we consider the chemical reaction $mX+nYarrow lZ$ andstands for the concentration of $X$ (resp.

$Y$$\tilde{u}$(resp.

$\tilde{v}$)).

The initial-boundary value problem $(1.1)-(1.3)$ was studied in [6] and it was proved
there that a unique solution $(u, v)(t, X)$ exists globally in time, this solution uniformly

converges to the equilibrium staterium state becomes $(0,0)$ when$\overline{u}_{0}=\overline{v}_{0}$), when$(\overline{u}_{0}-\overline{v}_{0},0)$with exponential rate and whenconstants

and If such that

$T$

as

$(u_{\infty}, 0)=(\overline{u}_{0^{-\overline{v}_{0},0)}}$$tarrow\infty$(note that this equilib-

and $n=1,$ $(u, v)(t, x)$ approaches$\overline{u}_{0}>\overline{v}_{0}$and $n>1$ , there exist positive$\overline{u}_{0}>\overline{v}_{0}$$||(u, v)(t)-(\overline{u}_{0}-\overline{v}0,0)||L\infty(\Omega)^{2}\leq K(1+t-T)^{-\alpha}$,$t\geq T$,

where $\alpha=1/(n-1)$ .
In [5], Hoshino and Kawashima studied more detailed large-time behavior of the solution $(u, v)(t, x)$ mentioned above. Concerning rate of convergence, they have shown thefollowing: Whenas$tarrow\infty$$\overline{u}0=\overline{v}_{0},$$(u, v)(t, x)$ converges to $(0,0)$ at the rate$t^{-\beta},$$\beta=1/(m+n-1)$ ,(Theorem 1).

We see that the polynomial rates of convergence stated above are just the same as thosefor the solution $(U, V)(t)$ to the corresponding system of ordinary differential equations(1.7)$\frac{dU}{dt}=-U^{m}V^{n}$,$\frac{dV}{dt}=-U^{m}V^{n}$

,

$t>0$

with the averaged initial data
(1.8)$(U, V)(0)=(\overline{u}_{0}, \overline{v}_{0})$.

This suggests the possibility that our solution $(u, v)(t, x)$ might behave like the solution$(U, V)(t)$ to the problem (1.7), (1.8) as. They have proved in [5] that the $(U, V)(t)$$tarrow\infty$

3

becomes an asymptotic solution for $tarrow\infty$ to the problem (1.1)$\overline{u}_{0}>\overline{v}_{0}$$\alpha=1/(n-1)$ ,$(u, v)(t, x)=(U, V)(t)+o(t^{-\alpha-1})$ ,$tarrow\infty$, while in the case where$\overline{u}0=\overline{v}_{0}$,$(u, v)(t, x)=(U, V)(t)+O(t^{-\beta 1}-)$ ,(1.10)

as(1.3). In fact, whenand $n>1$ ,

(1.9)

as

-

$tarrow\infty$$\beta=1/(m+n-1)$ ,(Theorem 2).

Moreover, they have also shown that the following asymptotic relations hold true:When$\overline{u}_{0}>\overline{v}_{0}$and $n>1$ ,(1.11)$(u, v)(t, x)=(\overline{u},\overline{v})(t)+o(t\mu-\alpha e-d0\lambda t)$

unifornly in$x\in\Omega$as(1.12)$-\triangle$

and when$\overline{u}_{0}=\overline{v}_{0}$,$(u, v)(t, X)=(\overline{u}, \overline{v}(t)+O(t^{\nu-\beta}e-d0\lambda t)$, where $d_{0}= \min\{d_{1}, d2\},$ is the smallest positive eigenvalue$\alpha=1/(n-1),$
with homogeneous Neumann boundary condition on

uniformly inof$tarrow\infty$$x\in\Omega$as$tarrow\infty$$\lambda$$\beta=$$\partial\Omega,$$1/(m+n-1),$ $\mu=k/(n-1)$ with $k=1+((\sqrt{2}-1)/2)n$ and $\nu=l/(m+n-1)$ with$\ell=1+(\sqrt{2(m^{2}+n^{2})}-(m+n))/2$ . Exponential decay estimates similar to above ones
are obtained in e.g., [2] and [8].
Our main results stated above are essentially based on a simple energy method which-estimates
makes use of the Poincar\’e inequality. Namely, we derive fundamental$L^{2}(\Omega)$by an energy method and then prove$L^{\infty}(\Omega)$-estimates by applying what we call$L^{p_{-}}L^{q}$estimate for the semigroup associated with the problem $(1.1)-(1.3)$ .
The plan of this report is as follows. In Section 2 we give precise statements of ourmain theorems. In Section 3 we discuss rates of convergence of the solution toward theequilibrium. We prove the large-time approximation results (1.9) (1.12) in the final-

section.
We will only give outlines of our discussion in this report. For the details, refer thereader to [5]. Furthermore, Hoshino [4] has shown that the argument in [5] is valid for adifferent reaction-diffusion system with homogeneous Neumann boundary conditions.

4

2. Main Results
In this section, we state our main results on the initial-boundary value problem of thereaction-diffusion system $(1.1)-(1.3)$ . Throughout the report we assume (1.4), (1.5) and$\overline{u}_{0}\underline{>}\overline{v}_{0}>0,$

where$\overline{u}_{0}$and$\overline{v}_{0}$are given in (1.6). The following results are proved in[5]: Under the assumptions stated above, the initial-boundary value problem $(1.1)-(1.3)$

has a unique global solution $(u, v)(t, x)$ which is smooth for $t>0$ . This solution verifies
the estimates $0\leq u(t, x)\leq||u_{0}||_{L(\Omega)}\infty$ andconverges to the equilibrium$(\overline{u}_{0}-\overline{v}0,0)$$0\leq v(t, x)\leq||v_{0}||_{L^{\infty}}(\Omega)$

uniformly in$x\in\Omega$asfor $t\geq 0,$$tarrow\infty$$x\in\Omega$and, that is,.

$\lim_{tarrow\infty}||(u, v)(t)-(\overline{u}_{0}-\overline{v}_{0,)|}0|_{L}\infty(\Omega)^{2}=0$

In order that we state our results, we define$u_{\infty}= \overline{u}_{0}-\overline{v}0=\frac{1}{|\Omega|}\int_{\Omega}(u_{0}-v0)(X)dx$

and
(2.1)

$\overline{u}(t)=\frac{1}{|\Omega|}\int_{\Omega}u(t, X)d_{X}$Integrate (1.1) over$(0, t)\cross\Omega$,$\overline{u}_{0}>\overline{v}_{0}$(2.2)
and when$\overline{u}(t)-\overline{v}(t)=\overline{u}_{0}-\overline{v}0$

for $t\geq 0$ . In particular,,$\overline{u}(t)=u_{\infty}+\overline{v}(t)$$\overline{u}_{0}=\overline{v}_{0}$.

and apply the Green formula to the resulting expression.

Then, it follows from (1.2) and (1.3) that

we have: When$\overline{v}(i)=\frac{1}{|\Omega|}\int_{\Omega}v(t, x)dx$,$t\geq 0$

,

,

(2.3)

$\overline{u}(t)=\overline{v}(t)$,$t\geq 0$.

Our first theorem, which is on rates of the above convergence as $tarrow\infty$ , can now be
stated as follows.

Theorem 1.

There exists a positive onstant$(1.1)-(1.3)$ satisfies the followin$\mathrm{c}$$gp\mathrm{r}$operties:$Ts\mathrm{u}\mathrm{C}h$that the solu tion $(u, v)(t, x)$ for5(i) When$\overline{u}_{0}>\overline{v}_{0}$and $n>1$ ,(2.4)$||(u, v)(t)-(u0\infty’)||L\infty(\Omega)^{2}\leq K(1+t-\tau)^{-\alpha}$(2.5)$||(u-\overline{u}, v-\overline{v})(t)||L^{\infty()}\Omega 2\leq K(1+t-\tau)K-d_{0}\lambda(t-\tau)e$for $t\geq T$ , where $\alpha=1/(n-1),$ $d_{0}= \min\{d_{1}, d_{2}\},$$of-\triangle$,$\lambda$

is the smallest positive eigenval ue

with homogeneo us Neumann $bo$ undary condition on

depending on(ii) When$||(u_{0}, v_{0})||_{L^{\infty()}}\Omega 2$$\overline{u}_{0}=\overline{v}_{0}$but not on$T$$\partial\Omega$

and $K$ denot es a constant

.

,
,

(2.6)

$||(u, v)(t)||L^{\infty()}\Omega 2\leq K(1+t-T)^{-\beta}$(2.7)$||(u-\overline{u}, v-\overline{v})(t)||L\infty(\Omega)^{2}\leq K(1+t-\tau)Ke^{-d\lambda}0(t-T)$

for $t\geq T$ , where $\beta=1/(m+n-1)$ and$d_{0},$$\lambda$

and $K$ are the same as in (i).

Our second theorem gives alarge-time approximation of the solution $(u, v)(t, x)$ . Before
stating the results, we summarize basic properties of the solution $(U, V)(t)$ to the problem(1.7), (1.8) for the corresponding system of ordinary differential equations. When$\overline{u}_{0}>\overline{v}_{0}$and $n>1$ , we have
(2.8)

$U(t)=u_{\infty}+V(t)$(2.9)$V(t)\sim t^{-\alpha}$

where $\alpha=1/(n-1).$ When$\overline{u}_{0}=\overline{v}_{0}$as,$t\geq 0$$tarrow\infty$,,, as the counterparts of (2.8) and (2.9), we have

(2.10)$U(t)=V(t)$ ,(2.11)$V(t)\sim t^{-\beta}$$t\geq 0$

as$tarrow\infty$,,

where $\beta=1/(m+n-1)$ . In fact, in this case, $V(t)=U(i)$ satisfies(2.12)$\frac{dV}{dt}=-V^{m+n}$, $t>0$and$V(0)=\overline{v}_{0}$6and is given explicitly as(2.13)$V(t)=\overline{v}_{0}\{1+(m+n-1)\overline{v}0t\}m+n-1-1/(m+n-1)$.

Large-time behavior of the solution $(u, v)(t, x)$ can now be described in terms of the$(U, V)(t)$ as follows.

Theorem 2.

Let $(u, v)(t, x)$ be the solu tion for (1.1)-(1.3) and let $(U, V)(t)$ be th$\mathrm{e}$solu tion to the problem (1.7), (1.8). Then the following asymptotic relations hold:(i) When

and $n>1$ ,$\overline{u}_{0}>\overline{v}_{0}$(2.14)$(u, v)(t, x)=(U, V)(t)+o(t^{-\alpha-1})$ ,(2.15)$(u, v)(t, x)=(\overline{u}, \overline{v})(t)+O(t^{\mu-a-}e)d0\lambda t$$\mathrm{u}\dot{m}f_{or}Idy$$\alpha,$$d_{0}$

and,in$\lambda$$x\in\Omega$

as $tarrow\infty$ . Here $(U, V)(t)$ and$(\overline{u}, \overline{v})(t)$verify (2.8), (2.9) and (2.2);

are the same as in Theorem 1, and $\mu=k/(n-1)$ with
$k=1+((\sqrt{2}-1)/2)n$ .

(2.16)(ii) When$\overline{u}_{0}=\overline{v}_{0}$,(2.17)$(u, v)(t, x)=(U, V)(t)+o(t^{-\rho_{-}1})$ ,(2.18)$(u, v)(t, x)=(\overline{u}, \overline{v})(t)+O(t\nu-\beta e-d0\lambda t)$

uniformly in$\beta,$$d_{0}$

and(2.19)

Remark.

$\lambda$$x\in\Omega$as$tarrow\infty$. Here $(U, V)(t)$ and$(\overline{u}, \overline{v})(t)$,verify (2.10), (2.11) and (2.3);

are the same as in Theorem 1, and $\nu=\ell/(m+n-1)$ with$\ell=1+(\sqrt{2(m^{2}+n^{2})}-(m+n))/2$ .

(i) Since $k>1$ in (2.16), we have $\mu>\alpha$ so that (2.15) actually involves

7

the polynomial growth order$t^{\mu-\alpha}$. Similarly, we have$\ell\geq 1$

in (2.19) and hence$\nu\geq\beta$.

However, the equality $\nu=\beta$ holds if and only if $m=n$ , and we can simplify (2.18) as$(u, v)(t, x)=(\overline{u}, \overline{v})(t)+O(e^{-d_{0}\lambda t})$in this case. This would be the optimal estimate.
(ii) If we put $m=0$ formally in (2.19), then the resulting$\ell$

coincides with the$k$in (2.16).

3. Rate of Convergence
In this section, we give an outline of proof of Theorem 1. We only discuss the casewhere$\overline{u}_{0}=\overline{v}_{0}$and show the estimates (2.6) and (2.7) in a series of lemmas. For (2.4)(resp. (2.5)) refer to [6] (resp. [5]).
First, we prove the$L^{2}(\Omega)$-decay estimate by a simple energy method which makes useof the following Poincar\’e inequality: For $w\in W^{1,2}(\Omega)$ satisfying(3. 1)

where$\lambda||w-\overline{w}||_{L(}^{2}2\Omega)\leq||\nabla w||^{2}L2(\Omega)$$\lambda$

is the smallest positive eigenvalue

condition on$\partial\Omega,$and$\overline{w}$$\overline{w}=\frac{1}{|\Omega|}\int_{\Omega}w(x)dx$

Let$\overline{u}_{0}=\overline{v}_{0}$on$\partial\Omega$,’with homogeneous Neumann boundaryis the mean value of $w(x)$ over(3.2)

Lemma 3.1.

$\mathrm{o}\mathrm{f}-\triangle$$\partial w/\partial\nu=0$$\Omega$, that is,.

. Then there exists a positive constant$T$(which is determin $ed$by (3.4) below) such that for $1\leq p\leq 2$,(3.3)$||(u, v)(t)||L\mathrm{p}(\Omega)^{2}\leq K(1+t-T)^{-\beta}$,

where $\beta=1/(m+n-1)$ and $K$ is a constant depending on

Outline of proof.
positive constant
(3.4)

$T$

$t\geq T$,$||(u_{0}, v_{0})||_{L(}\infty\Omega)2$

but not on$T$.

It suffices to prove (3.3) for $p=2$ . Note that we can choose aso large that$||(u, v)(i)||_{L\infty}^{m}+n_{\Omega}-)(21\leq d_{0}\lambda/C$

for $t\geq T$.

8

By the simple energy method, the Poincar\’e inequality (3.1) and the H\"older inequality, we

can get$\frac{1}{2}\frac{d}{dt}||(u, v)(t)||2L2(\Omega)^{2}+C||(u, v)(t)||L^{2}(\Omega)^{2}m+n+1\leq 0$

which leads us to (3.3) with $p=2$ .
Next we show the$L^{\infty}(\Omega)^{2}$,$t>T$ ,$\square$-decay estimate (2.6). We will use thesemigroup associated with the heat equation. Let us denote byhomogeneous Neumann boundary condition on$\partial\Omega$

and let$L^{p_{-}}L^{q}$estimate for thethe operator$A$$\{e^{-tA}\}_{t\geq 0}$$-\triangle$

withbe the corresponding

semigroup. For given $w(x)$ , we define(3.5)$P_{0}w= \frac{1}{|\Omega|}\int_{\Omega}w(X)d_{X}$(3.6)$(P_{+}w)(x)=w(x)-P0w$ .

Note that$P_{0}w$

is just the mean value$\overline{w}$defined in (3.2). It is well known thatare the projections onto the eigenspaces of$\lambda=0$,$A$$P_{0}$

and$P_{+}$

corresponding to the principal eigenvalue

and to the totality of positive eigenvalues, respectively. It is also known that thesemigroup$e^{-tA}$(3.7)

satisfies the following$L^{p_{-}}L^{q}$estimate: For $1\leq q\leq p\leq\infty$ ,$||e^{-tA}P+w||_{L^{\mathrm{p}}}(\Omega)\leq Cm(t)^{-}(N/2)(1/q-1/p)\lambda t|e^{-}|P_{+}w||_{L^{q}}(\Omega)$

where $m(t)= \min\{1,t\},$(3.1) and$)$$C$$\lambda$

is the smalest positive eigenvalue of$A$,(the same as the$\lambda$

inis some positive constant. For the details of (3.14), see [1], [3] and [7].

By means of the$L^{p_{-}}L^{q}$estimate stated above, we can prove the$L^{\infty}(\Omega)^{2}$-decay estimate(2.6).

Lemma 3.2.
where

$T$

Let

$\overline{u}_{0}=\overline{v}_{0}$. Then the$L^{\infty}(\Omega)$-decay estim ate (2.6) holds true for $t\geq T$ ,is the constant in Lemma 3.1 and is determin $ed$ by (3.4).

Outline of proof.

We make use of the decomposition $w=P_{0}w+P_{+}w$ . Theof the solution is estimated by using$|P_{0}w|\leq|\Omega|^{-1}||W||L^{1}(\Omega)$$|P_{0}(u, v)(t)|\leq K(1+t-T)^{-\beta}$,$P_{0}$and (3.3) with $p=1$ as$t\geq T$

,

-part

9

where $\beta=1/(m+n-1)$ . Here and in what follows $K$ denotes a constant depending on$||(u_{0}, v_{0})||L^{\infty}(\Omega)^{2}$

but not on$T$.

It remains to prove(3.8)$||P_{+}(u, v)(t)||L^{\infty}(\Omega)^{2}\leq K(1+t-T)^{-\beta}$with the same . To this end we use the semigroup$\beta$$e^{-td_{1}A}$,$i\geq T$,

and transform the first equationof (1.1) into an integral equation. After applying the projection

defined by (3.6), we$P_{+}$

obtain$(P_{+}u)(t)=e^{-(t-\tau}()d_{1}AP_{+}u)(T)- \int_{T}^{t}e^{-(-}Pt\mathcal{T})d1A+(umv^{n})(\mathcal{T})d_{\mathcal{T}}$We have a similar equation also for , in which$v$$d_{1}$

is replaced by$d_{2}$,$t\geq T$.

. If we apply the$L^{p_{-}}L^{q}$estimate (3.7) to these integral equations, then we get(3.9)$||P_{+}(u, v)(t)||L\mathrm{p}(\Omega)^{2}\leq Ke^{-d\mathrm{o}\lambda(tT)}-$$+K \int_{T}^{t}m(t-\tau)^{-}(N/2)(1/q-1/p)-d0\lambda(t-\tau)|e|P+(u, V)(\tau)||Lq(\Omega)^{2}d\mathcal{T}$where $1\leq q\leq p\leq\infty$ . Here we used the fact that

$P_{+}(u^{m}v^{n})$,$t\geq T$,

satisfies$||P+(u^{m}v)n(t)||L^{\mathrm{p}}(\Omega)$$\leq C(||u(t)||_{L^{\infty}}^{m-}(\Omega)||v(t)||^{n}L\infty(\Omega)1||P+^{u}(t)||_{L}\mathrm{p}(\Omega)+||u(t)||_{L^{\infty}}^{m}(\Omega)||v(t)||_{L^{-1}(}^{n}\infty\Omega)||P_{+}v(t)||_{L^{\mathrm{p}}(}\Omega))$or, in a more compact form,(3.10)$||P_{+}(uv^{n}m)(t)||L\mathrm{p}(\Omega)\leq C||(u, v)(t)||L^{+}m\infty(\Omega)n-12||P_{+}(u, v)(t)||L\mathrm{p}(\Omega)^{2}$,

where $1\leq p\leq\infty$ . We require$(N/2)(1/q-1/p)<1$ .

(3.11)We use (3.9) for suitable

$p$

and

$q$

to prove the desired estimate (3.8). When $N=1,2$

or 3, we put $p=\infty$ and $q=2$ in (3.9). When $N\geq 4$ , we define$1/p_{j}-1/p_{j+1}=1/N,$ $j=0,1,2,$ $\cdots,$ $N-1$ .

We see that$\{p_{j}\}$such that $p_{N-1}<\infty$ and $p_{N}=\infty$ . We now put $p=p_{j+1}$ and(3.11), we can show (3.8).

$\square$$\{p_{j}\}$by $p_{0}=1$ andis an increasing sequence$q=p_{\mathrm{j}}$in (3.9) which satisfy

10Once the(2.7) rather$L^{\infty}(\Omega)^{2}$$\mathrm{e}\mathrm{a}s$-decay estimate (2.6) is known, one can prove the as ymptotic relation

ily as follows.

Lemma 3.3.

Let$\overline{u}_{0}=\overline{v}_{0}$. Then (2.7) holds true for $t\geq T$ , where$T$

is th constant in$\mathrm{e}$Lemma 3.1.

Outline of $proofl$.

It follows from (3.10) and (2.6) that$||P_{+}(u, v)(t)||L^{\mathrm{p}}(\Omega)2$$\leq$$Ce^{-d_{0}\lambda}(t-\tau)||P_{+}(u, v)(\tau)||L\mathrm{p}(\Omega)^{2}$$+I \mathrm{f}\int_{T}^{t}e^{-d}-)(0\lambda(t\mathcal{T}1+\tau-T)-1||P_{+}(u, v)(\tau)||L\mathrm{p}(\Omega)^{2}d_{\mathcal{T}}$for $t\geq T$ . Then Gronwall’s lemma yields$||P_{+}(u, v)(t)||L\mathrm{p}(\Omega)^{2}\leq C||P_{+}(u, v)(\tau)||L^{\mathrm{p}}(\Omega)^{2}(1+t-T)^{K}e-d0\lambda(t-\tau)$,$t\geq T$.

$\square$

4. Large-time approximation

The aim of this section is to prove Theorem 2. We only discuss the case where
and $n>1$ can be studied along the similar manner.
also here. The case where$\overline{u}_{0}=\overline{v}_{0}$$\overline{u}_{0}>\overline{v}_{0}$We first give an outline of proof of the asymptotic relations (2.17) and (2.18) under theis smaladditional restriction that the initial perturbation from the mean value (enough in Subsection 4.1. Then in Subsection 4.2 we remove this restriction imposed on$\overline{u}_{0},$the initial data and prove (2.17) and (2.18) in$\mathrm{f}\mathrm{u}\mathrm{U}$$\overline{v}_{0)}$generality. For the details of (2.18) and(2.19), refer to [5].

4.1. The case$\overline{u}_{0}=\overline{v}_{\mathrm{o}}$with small initial data

Let us consider the case where$\overline{u}0=\overline{v}_{0}$. We want to estimate the solution $(u, v)(t, x)$

to the problem $(1.1)-(1.3)$ in the form
(4. 1)$u(t, x)=V(t)(1+\phi(t, x)),$$v(t, x)=V(t)(1+\psi(t, X))$ ,where $V(t)$ is the solution of (2.12) and is given explicitly as in (2.13). A straightforward

11

computation shows that the $(\phi, \psi)(t, x)$ defined by (4.1) must satisfy(4.2)$\phi_{t}=d_{1}\triangle\phi-V(t)m+n-1\{(m-1)\phi+n\psi_{+}f\}$,$t>0,$$x\in\Omega$,$\psi_{t}=d_{2}\Delta\psi_{-}V(t)^{m}+n-1\{m\phi+(n-1)\psi+f\}$,$t>0,$$x\in\Omega$

,

$\{$

(4.3)

$\frac{\partial\phi}{\partial\nu}=\frac{\partial\psi}{\partial\nu}=0$,$t>0,$$(\phi, \psi)(0, X)=(\phi_{0}, \psi 0)(X)$,$x\in\partial\Omega$$x\in\Omega$

,

,

where
$f=(1+\phi)^{m}(1+\psi)^{n}-(1+m\phi+n\psi)$ ,(4.4)$\phi_{0}(_{X})=(u_{0}(x)-\overline{u}_{0})/\overline{v}_{0}$Note that we used$\overline{u}_{0}=\overline{v}_{0}$namely, the mean value of,$\psi_{0}(x)=(v_{0}(X)-\overline{v}0)/\overline{v}_{0}$.

in the first relation of (4.4). Note also that $P_{0}\phi 0=P_{0}\psi 0=0$ ,$(\phi_{0}, \psi_{0})(x)$vanishes.

We will prove the following theorem for $(\phi, \psi)(t, x)$ that implies (ii) of Theorem 2 inthe case where the initial perturbation from the mean value is small enough.

. Then there exists a positive constantdefin $ed$ by (4.1) satisfies, then theLet

Theorem 4.1.

$||(\phi 0, \psi 0)||L\infty(\Omega)^{2}\leq\epsilon_{0}$$\overline{u}_{0}=\overline{v}_{0}$,$||(\phi, \psi)(t)||L^{\infty}(\Omega)^{2}\leq C||(\phi 0, \psi 0)||_{L^{\infty}(\Omega)}2(1+t)^{-}1$(4.6)$||P_{+}(\phi, \psi)(t)||L\infty(\Omega)^{2}\leq C||(\phi_{0}, \psi_{0})||L\infty(\Omega)2(1+t)^{\nu}e^{-d_{0}}\lambda t$$d_{0}= \min\{d_{1}, d_{2}\},$$C$$\lambda$such that if$(\phi, \psi)(t, x)$(4.5)

for $t\geq 0$ , where$\epsilon_{0}$is a constant and$d_{0},$$\lambda$

and

$\nu$

are the same as in Theorem 2, that is,is the smallest positive eigenvalue of the operator A defined in Se ction

3 and $\nu=\ell/(m+n-1)$ with$\ell$in (2.19).

12This theorem will be proved in a series of lemmas below. We use the following notations:
For $1\leq p\leq\infty$ ,$=||(\phi_{0}, \psi_{0})||L\mathrm{p}(\Omega)^{2}$$I_{p}$$M_{p}(t)$,$= \sup_{0\leq\tau\leq t}(1+\tau)||(\phi, \psi)(\tau)||L\mathrm{p}(\Omega)^{2}$$M_{\infty}^{0}(t)$$= \sup_{0\leq\tau\leq t}(1\dagger\tau)|P_{0}(\phi, \psi)(_{\mathcal{T})}|$$M_{p}^{+}(t)$,,$= \sup_{0\leq\tau\leq t}(1+\tau)-\nu e|d0\lambda\tau|P_{+}(\phi, \psi)(\tau)||L\mathrm{p}(\Omega)^{2}$.

Obviously we have$I_{q}\leq CI_{p}$,$M_{q}(t)\leq CM_{p}(t)$,$M_{q}^{+}(t)\leq CM_{p}^{+}(t)$for $1\leq q\leq p\leq\infty$ , where $C=|\Omega|^{1/q-1}/p$ .
We state some lemmas which are used in order to prove Theorem 4.1 without proofs.

First, we estimate theLemma 4.2.

$P_{0}$-part of the $(\phi, \psi)(t, x)$ .

It holds true that

(4.7)

$M_{\infty}^{0}(t)\leq K(t)M_{2}(t)^{2}$,$t\geq 0$,

where and in what follows $K(t)$ denotes a quantity $dep$ endin only on$g$$\sup_{0\leq\tau\leq t}||(\phi, \psi)(\mathcal{T})||L^{\infty}(\Omega)^{2}$This lemma can be shown with use of (1.1), (3.5) and the facts that $P_{0}\phi_{0}=P_{0}\psi_{0}=0$
and $P_{0}\phi(t)=P_{0}\psi(t)$ for all $t\geq 0$ .
Second, we estimate theLemma 4.3.

$P_{+}$-part of the$(\phi, \psi)(t, x)$

by using an energy method.

For $1\leq p\leq 2_{2}$(4.16)

Third, we derive theintegral equations which$M_{p}^{+}(t)\leq CI_{2}+K(t)M_{\infty}(t)M^{+}2(t)$$L^{\infty}(\Omega)^{2}$$(\phi, \psi)$-estimate by applying themust satisfy.

.

$L^{p_{-}}L^{q}$estimate (3.7) to the.

13

The following estimate holds true:

Lemma 4.4.

(4.8)

$M_{\infty}^{+}(t)\leq CI_{\infty}+K(t)M_{\infty}(t)M+(\infty t)$.

We can complete the proof of Theorem 4.1 asOutfine of proof of Theorem 4.1.
follows. By definition we see that
(4.9)

$M_{\infty}(t)\leq C(M_{\infty}^{0}(t)+M_{\infty}^{+}(t))$

for some constant(4.10)$C$. Therefore, combining (4.7) and (4.8), we have

$M_{\infty}^{0}(t)+M_{\infty}^{+}(t)\leq CI_{\infty}+K(t)(M^{0}(\infty)t+M_{\infty}^{+}(t))^{2}$

Recall that $K(t)$ depends only on.

$\sup||(\phi, \psi)(\tau)||_{L^{\infty}(}\Omega)2$.

and hence is considered as a$0\leq\tau\leq t$function of$M_{\infty}(t)$inequality for

or of$M_{\infty}^{0}(t)+M_{\infty}^{+}(t)$$M_{\infty}^{0}(t)+M_{\infty}^{+}(t)$by (4.9). Therefore (4.10) can be regarded as an

and is solved in the form

$M_{\infty}^{0}(t)+M_{\infty}^{+}(t)\leq CI_{\infty}$

provided that

$I_{\infty}$

is suitably small, say$I_{\infty}\leq\epsilon_{0}$,$t\geq 0$,. Consequently, we have

$M_{\infty}(t)\leq CI_{\infty}$

andfor $t\geq 0$ , which imply the desired estimates (4.5) and (4.6), respectively.

$M_{\infty}^{+}(t)\leq CI_{\infty}$

This completes the proof of Theorem 4.1.

4.2. The case$\overline{u}_{\mathrm{O}}=\overline{v}_{\mathrm{o}}$$\square$with large initial data

In the previous subsection, we have proved (ii) of Theorem 2 for initial data close to
the mean value. The aim of this subsection is to remove that restriction imposed on theinitial data. To this end we first observe the following asymptotic relation.
Lemma 4.5.

$L\mathrm{e}i\overline{u}_{0}--\overline{v}_{0}$and let $(u, v)(t, x)$ be the solution for $(1.1)-(1.3)$ . Then it

holds true that$||(u-\overline{u}, v-\overline{v})(t)||L^{\infty()}\Omega 2/\overline{v}(t)arrow 0$

where$(\overline{u},\overline{v})(t)$as$tarrow\infty$,

is the mean value of $(u, v)(t, x)$ and is defined by (2.1); note

for $t\geq 0$ by (2.3).

$t\Lambda at\overline{u}(t)=\overline{v}(t)$14This lemma can be proved by the comparison theorem and (2.7).
By virtue of Lemma 4.5, we can apply Theorem 4.1 for large time and prove (ii) of

Theorem 2 without any restriction on the size of the initial data.

Outfine of proof of (ii) of Theorem 2.

Let$\epsilon_{0}$be the positive constant in Theorem 4.1.

By virtue of Lemma 4.5, we can choose a constant $T_{0}\geq T$ ((4.11)

for$||(u-\overline{u}, v-\overline{v})(i)||L\infty(\Omega)^{2}/\overline{v}(t)\leq\epsilon_{0}$For this choice of$T_{0}$, we define$(\tilde{U},\tilde{V})(t)$$\frac{d\tilde{U}}{di}=-\tilde{U}^{m}\tilde{V}^{n}$,for$t\geq T_{0}$in Lemma 3.1) so large that$t\geq T_{0}$.

as the solution to the problem$\frac{d\tilde{V}}{dt}=-\tilde{U}^{m}\tilde{V}^{n}$,$(\tilde{U},\tilde{V})(T_{0})=(\overline{u},\overline{v})(\tau_{0})$$\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{e}\overline{u}(\tau_{0})=\overline{v}(T_{0})$$T$$t>T_{0}$,.

by (2.3), we see that$\tilde{U}(i)=\tilde{V}(t)$(4.12),$t\geq T_{0}$as$\tilde{V}(t)\sim t^{-\beta}$

where $\beta=1/(m+n-1)$ , and that$\tilde{V}(t)$$\frac{d\tilde{V}}{dl}=-\tilde{V}^{m+n}$,,$tarrow\infty$,

solves$t>T_{0}$,$\tilde{V}(T\mathrm{o})=\overline{v}(T_{0})$.

Now, as in the previous subsection, we estimate the solution $(u, v)(t, x)$ to the problem$(1.1)-(1.3)$ in the form$u(t, x)=\tilde{V}(t)(1+\tilde{\phi}(t, x))$$(\tilde{\phi},\tilde{\psi})(t, x)$$\tilde{V}(t)$. Moreover, we have

(4.13).

$v(t, x)=\tilde{V}(t)(1+\tilde{\psi}(t, X))$

introduced here satisfies (4.2) and (4.3) (for $t>T_{0}$ ) with $V(t)$ replaced byThe

be applied to,$(\tilde{\phi},\tilde{\psi})(t, x)$$||(\tilde{\phi},\tilde{\psi})(T_{0})||L^{\infty}(\Omega)^{2}\leq\epsilon_{0}$for$t\geq T_{0}$by (4.11). Therefore Theorem 4.1 canand we obtain,$||(\tilde{\phi},\tilde{\psi})(t)||_{L}\infty(\Omega)^{2}\leq C||(\tilde{\phi},\tilde{\psi})(T_{0})||_{L(}\infty\Omega)^{2}(1+t-T_{0})^{-1}$15(4.14)
for$t\geq T_{0}$$||P_{+}(\tilde{\phi},\tilde{\psi})(t)||L\infty(\Omega)^{2}\leq C||(\tilde{\phi},\tilde{\psi})(T\mathrm{o})||_{L^{\infty(\Omega}})2(1+t-\tau 0)^{\nu}e^{-d_{0}\lambda}(t-^{\tau_{0})}$. The estimates (4.12) and (4.14) applied to the expression$(u, v)(t, x)=(\overline{u}, \overline{v})(t)+\tilde{V}(t)(P_{+}\tilde{\phi}, P_{+}\tilde{\psi})(t, x)$give the desired asymptotic relation (2.18). In order that we prove (2.17), we write$u(t, x)=U(t)+\tilde{V}(t)(1-V(t)/\tilde{V}(t)+\tilde{\phi}(t, x))$,$v(t, x)=V(t)+\tilde{V}(i)(1-V(t)/\tilde{V}(t)+\tilde{\psi}(t, X))$,

where $V(t)$ is the solution of (2.12) and where we used (2.10). Since we already verified(4.12) and (4.13) and we can show the following lemma that concerns with the difference

between $V(t)$ andLemma 4.6.

$\tilde{V}(t)$

When

for

$tarrow\infty$$\overline{u}_{0}=\overline{v}_{0}$, we can obtain (2.17).

, we have

V $(t)/\tilde{V}(t)-1=O(t^{-1})$as$tarrow\infty$.

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Anal. Appl., 118 (1986), 455-466.
[3] D. Henry, “Geometric Theory of Semilinear Parabolic Equations”, Lecture Notes in

Math. 840, Springer-Verlag, Berlin, 1981.
[4] H. Hoshino, Rate of convergence of global solutions for a dass of reaction-diffusion
systems and th e corresponding asymptotic solutions, to appear in Adv. Math. Sci.

Appl.

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[5] H. Hoshino and S. Kawashima, Asymptotic equivalence of a reaction-diffu sion system

to the corresponding system of ordinary differ ential eq uation s, to appear in Math.

Models Meth. Appl. Sci.
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Department of Applied Mathematics

Fukuoka University

Nanakuma 8-19-1, Jonan-ku
Fukuoka 814-80, Japan〒 814-80 福岡県福岡市城南区七隈 8-19-1

福岡大学理学部応用数学教室