数理解析研究所講究録
第 1704 巻 2010 年 195-202

195

Existence of multiple stable stationary patterns tosome reaction-diffusion equation in heterogeneousenvironments
首都大学東京大学院理工学研究科 倉田 和浩 (Kazuhiro Kurata)
Department of Mathematics and Information Sciences,
Tokyo Metropolitan University

1

Introduction and main results

In this paper we report some recent our mathematical research on the existenceof multiple non-trivial stable stationary patterns for some rection-diffusion euquations
with heterogeneous environments.
We emphasize that the existence of multiple stable stational patterns also depends not
only on heterogenous environments but also on the nonlinearity. In fact, for the logistic
model:$0=d\triangle u+u(b(x)-u)$ ,$x\in\Omega$,$\partial u_{=0}$$\overline{\partial n}$,$x\in\partial\Omega$

,

where
is a bounded smooth domain, this phenomenon does not occur. It is
well-known that a positive solution of this boundary value problem must be unique
if exists. Actually, it exists if $(1/d)>\lambda_{1}(b)>0$ when $\int_{\Omega}bdx<0$ , any $d>0$ when
. Here,is the first positive eigenvalue$(x\in\Omega)$ ,. Asymptotic behaviour of the unique solution $u=u_{d}$ is aloswell-known as $darrow 0$ .
In this paper we consider two reaction-diffusion model with weak Allee effect, whichappears as a populationa growth model, and show existence of multiple stable stationary patterns under certain heterogeneous evnvironments. Throughout this paper, we
say that a solution is stable, which is called weakly stable sometimes, if it is a local
minimizers of the associated energy functional.
There are several works on the existence of stable staionary patterns for the rectiondiffusion model with strong Allee effect and environment factor $a(x)$ :$\Omega\subset R^{N}$$\int_{\Omega}bdx\geq 0$$\partial 0^{l}/\partial n=0,$$\lambda_{1}(b)$$of-\triangle\phi=\lambda b(x)\acute{o},$$(x\in\partial\Omega)$$\frac{\partial u}{\partial t}=d\triangle u+u(u-a(x))(1-u)$,

where $0<a(x)<1$ (see, e.g. Dancer-Yan([3]) and the references therein).

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2

Problem l(weak Allee Effect)

We consider the following stationary problem for a rection-diffusion model with weak
Allee effect under Neumann boundary condition:$0=d\Delta u+u(b(x)u-u^{2})$ ,$\frac{\partial u}{\partial n}=0$,$x\in\Omega$$x\in\partial\Omega$,,

is a bounded smooth domain. We consider the following situation:$b(x)=-1$ on. This means that a habitat
and $b(x)=1$ on.
and a bad region
consists of a good region
Question: How this heterogeneous environment has an influence on patterns ofstable positive solutions?
In [5], Ide, Kurata and Tanaka studied this problem and obtained the following
results.

where$\Omega\subset R^{N}$$\Omega=\Omega_{+}\cup\Omega_{-}$$\Omega_{-}$$\Omega_{+},$$\Omega$$\Omega_{-}$$\Omega_{+}$

Theorem 1 $(a)$ If
then for sufficiently small $d>0$ there exists at least
two type of positive solutions.
$(b)$ Assume $N=1$ andhas well-separated k-components. Then for sufficiiently
small $d>0$ , there exists at least $2^{k}-1$ types of stable positive solutions.
$\int_{\Omega}bdx<0_{f}$$\Omega_{+}$The precise condition on the meaning of the well-separated components, see [5]. In [8]we extended the part (b) of Theorem 1 (i.e. the existence of multiple stable patterns)in the higher dimensional case.
Theorem 2 Suppose $1\leq N\leq 3,$
has well-separated k-components. Thenfor sufficiently small $d>0$ , there exists at least $2^{k}-1$ types of stable positive solutions.
Actually, these stable solutions are local minimizers in.
$\overline{\Omega_{+}}\subset\Omega,$$\Omega_{+}$$H^{1}(\Omega)$. So it is sometomes
Throughout this paper, stability means local minimizers incalled weak stability and it is not clear whether the solutions obtained above have alinearized stability.
$H^{1}(\Omega)$

3

Problem 2(Balanced Bistable Nonlinearity)

We consider another rection diffusion model with weak Allee effect:$-\epsilon^{2}\triangle u=f(x, u(x))$

where$\epsilon>0,$$\Omega\subset R^{N},$$N\geq 1$,

in$\Omega$,$\frac{\partial u}{\partial n}=0$,on$\partial\Omega$

is a bounded smooth domain. Here$f(x, u)=a(x)|u|^{p-1}u-|u|^{q-1}u$ ,$1\leq p<q$ ,and $a(x)>0(x\in\Omega)$ . An associated energy functional is as follows:$J_{\epsilon}(u)= \frac{\epsilon^{2}}{2}\int_{\Omega}|\nabla u|^{2}dx-\int_{\Omega}F(x, u(x))dx$

,

,

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$F(x, u)= \int_{0}^{u}f(x, s)ds=\frac{1}{p+1}|u|^{p+1}a(x)-\frac{1}{q+1}|u|^{q+1}$.

It is well-known that solutions of the boundary value problem above corresponds tocritical points ofon. This model has a so-called balanced bistable nonlinearity: for fixed$\gamma=1/(q-p)$ , are stable states in ODE sense andthe $potential-F(x, u)$ takes its minimum same energy at two states. As atypical examples, we have as $p=1,$ $q=3$ ,$H^{1}(\Omega)$$J_{\epsilon}(u)$$x\in\Omega,$$u=\pm a(x)^{\gamma},$$u=\pm a(x)^{\gamma}$$-\epsilon^{2}\Delta u=u(x)a(x)-u(x)^{3}$,in$\Omega$,

which is called Allen-Cahn equation. When $p=2,$ $q=3$ ,.
Fo this model, it is rather easy to show that there exists a global minimizer
isand$-\epsilon^{2}\triangle u=|u(x)|u(x)a(x)-u(x)^{3}$in,$\Omega$$U_{\epsilon}(x)$which$U_{\epsilon}(x)>0(x\in\Omega)$$\sup_{x\in\Omega}|U_{\epsilon}(x)-a(x)^{\gamma}|arrow 0$

as. Moreover, global minimizers should not change sign.
Here we have a following question: Does there exist a stable solution (e.g. local minimizer)
such that$\epsilonarrow 0$$u_{\epsilon}(x)$$u_{\epsilon}(x)\sim a(x)^{\gamma}$$u_{e}(x)\sim-a(x)^{\gamma}$

for some

as

on
on

$\Omega^{+}$,$\Omega^{-}$? Namely, does there exist a stable stationary pattern? Inother word, does there exist a stable stationary sign-changing solution?
The following results are known:$\bullet$$\Omega^{\pm}\subset\Omega$$\epsilonarrow 0$(H.Matano [9]) Let $p=1,$ $q=3,$ $N\geq 1$ and $a(x)\equiv 1$ . Then, for any convex
domain , there are no non-constant local minimizer. However, for some nonconvex domain , there exists a sign-changing stable solution.
$\Omega$$\Omega$$\bullet$(K. Nakashima[ll]) Let $p=1,$ $q=3,$ $N=1$ and $a(x)$ has a non-degenerate
minimum at $x=r_{0}\in\Omega=(0,1)$ . Then there exists a stable solution $u(x)$ s.t.

on
and $u(x)\sim-\sqrt{a(x)}$ on.
In[6], Kurata and Matsuzawa studied this problem under the following situation: Let$u(x)\sim\sqrt{a(x)}$$(r_{0},1)$$(0, r_{0})$$p=I,$ $q=3,$ $N\geq 1$ . Suppose there exit $D_{j}(j=1,2)$ s.t.

$\overline{D_{j}}\subset\Omega(j=1,2)$$\overline{D_{1}}\cap\overline{D_{2}}=\emptyset,$

and$a(x)=1$ on $D=D_{1}\cup D_{2}$ ,$a(x)=O(\epsilon^{2})$on$\Omega\backslash \overline{D}$.

Then we obtained the following result.

Theorem 3 Under the assumptions above, there exists a local $minimizeru_{\epsilon}s.t$ .
$u(x)\sim-1$ on
on
and
on
.
In [8] (see also [13]), Kurata and Yanai studied this problem under the following
situation: Let $p=2,$ $q=3,$ $N\geq 1$ . Suppose there exit $D_{j}(j=1,2)$ s.t.

$u_{\epsilon}(x)\sim$$1$$D_{1},$$D_{2}$$|u(x)|\leq C\epsilon$$\Omega\backslash \overline{D}$$\overline{D_{1}}\cap\overline{D_{2}}=$$\emptyset,$$\overline{D_{j}}\subset\Omega(j=1,2)$

and$a(x)=1$ on $D=D_{1}\cup D_{2}$ ,on.
Theorem 4 Under the assumptions above, there exists a local $minimizeru_{\epsilon}s.t$ .
$u(x)\sim-1$ on
on
and
on
.
$1$

$D_{1},$$D_{2}$$|u(x)|\leq C\epsilon$$a(x)=O(\epsilon)$$\Omega\backslash \overline{D}$$\Omega\backslash \overline{D}$$u_{\epsilon}(x)\sim$

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4

Basic Strategy for the Proofs of Theorem 2 and4

In this section, we explain the basic strategy and the outline of the proof of Theorem 2
and the keypoints of the proof of Theorem 4. We omit the proof of Theorem 1 and 3.
For the details, see [5], [8], [6], [8], [13].
Our basic strategy is to use varitaional methods and a sub-supersolution method.
In the construction and estimates of a suitable sub-supersolution, we use boundary
blow-up solutions to the following problem:$V(x)>0$ ,$-\Delta V=g(V),$$V(x)=0(x\in\Gamma_{1})$

,

in

$G$

,

$V(x)=+\infty(x\in\Gamma_{2})$,.
may be an empty
where is a domain or an annular domain and$g(t)$
problemsuitably for eachset. We choose the nonlinear function
1 and 2.
We use Dancer-Yan’s compariosn lemma for minimizers.
$G$$\partial G=\Gamma_{1}\cup\Gamma_{2}$$\Gamma_{1}$Lemma 1 Let $h_{j}(t),j=1,2$ , be continuous functions $s.t$ . $h_{i}(t)>0(t\leq 0);h_{i}(t)\leq$$0(t\geq c)$ . Let $H_{j}(t)= \int_{0}^{t}h_{j}(s)ds,j=1,2$ and$J_{\lambda,j}(v)= \frac{1}{2}\int_{\Omega}|\nabla v|^{2}dx-\lambda\int_{\Omega}H_{j}(v)dx$.

Suppose $h_{1}(t)\geq h_{2}(t),$ $(0\leq t\leq c),$ $\eta_{j}(x)\in H^{1}(\Omega)\cap C$ (St), $j=1,2s.t$ . $c\geq\eta_{1}(x)\geq$,. Let $u_{j}(x),j=1,2$ be minimizers to
the minimizing problem:$\eta_{2}(x)\geq 0,$$(x\in\partial\Omega)$$\eta_{1}(x)\not\equiv\eta_{2}(x),$$\inf\{J_{\lambda,j}(v) :Then we have $u_{1}(x)\geq u_{2}(x),$$(x\in\Omega)$$(x\in\partial\Omega)$u(x)=\eta_{j}(x), (x\in\partial\Omega)\}$.

.

For the proof of Lemma 1, see [3] and [6].

4.1Outline of the proof of Theorem 2

Let $f(x, t)=t_{+}^{2}(b(x)-t)$ and $F(x, t)= \int_{0}^{t}f(x, s)ds$ . Let 1 $\leq N\leq 3$ . Define, for$u\in H^{1}(\Omega)$,$I(u)=I(u; \Omega)=\frac{d}{2}\int_{\Omega}|Du|^{2}dx-\int_{\Omega}F(x, u)dx$,

The following propostion is easy to show.
Proposition 1 Let $u=u_{d}$ be a global minimizer of $I(u;\Omega)$ on
to 1 as $darrow 0$ unifomly on any compact subset ofand tends toon any compact subset of .

. Thentends
as $darrow 0$ uniformly$H^{1}(\Omega)$$\Omega_{+}$$\Omega_{-}$$0$$u_{d}$

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Now, we explain how to construct non-trivial local minimizers.
For simplicity, assume is a one component of , and show how to construct a
local minimizer
which behaves 1 on
and onas $darrow 0$ . We also
assume the following conditions(i.e. is well-separated to other components of ):(A): There exists $R_{1}>r_{0}>0$ and $x_{0}\in A$ such that.
Furthermore,should be large enough which will be specified later. Here $B_{f}(y)=$$\{x||x-y|\leq r\}$ .
$A$$\Omega_{+}$$\tilde{u}_{d}$$\Omega_{-}\cup A$$0$$\Omega_{+}\backslash A$$A$$\Omega_{+}$$A\subset B_{r_{0}}(x_{0}),$$B_{R_{1}(xo)}\backslash$$B_{r0}(x_{0})\subset\Omega_{-}$$R_{1}$

4.2

Construction of Sub-solution

Take $R=2r_{0}+\delta$ with sufficiently small $\delta\in(0,1/8)$ , e.g. $\delta=1/16$ and fix . Thenshould be large enough to satisfy $R_{1}>16r_{0}+1$ and $C_{0}/R_{1}^{2}\leq v(R)$ , where $C_{0}>0$ isa certain universal constant and $v(R)>0$ is a constant depending only on $R,$ . Notethat this implies $R_{1}>8R$ .
Let
be a minimizer of the problem:$R$$R_{1}$$r_{0}$$\underline{v}_{d}$$\inf\{I(u;\Omega\backslash B_{R}(x_{0}))|u\in H^{1}(\Omega\backslash B_{R}(x_{0})),$

Now, define
to show

Lemma 2for$\underline{u}_{d}(x)=\underline{v}_{d}(x)$$\underline{u}_{d}$$x\in\Omega\backslash B_{R}(x_{0})$$u=0$ on$\partial B_{R}(x_{0})\}$.

, and $=0$ for $x\in B_{R}(x_{0})$ . Then it is easy

is a subsolution of the problem.

4.3

Construction of Super-solution

Define$\overline{b}(t)=1$

for $t\in[0, r_{0}]$ , and $=-1$ for $t\in[r_{0}, R]$ . Then$\int_{0}^{R}\overline{b}(t)dt=-R+2r_{0}=-\delta<0$.

Now, it is known that the ODE problem$-v”(t)=v^{2}(t)\overline{b}(t),$$0<t<R$ ,$v’(O)=v’(R)=0$

has a positive solution $v(t)>0(t\in[0, R])$ (this solution is a mountain pass typesolution)([l]). It is easy to see $v’(t)<0(0<t<R)$ .
Lemma 3 Let $v_{d}^{*}(x)=dv(|x-x_{0}|)$ . Then we have$-d\triangle v_{d}^{*}-f(x, v_{d}^{*})\geq 0$,$x\in B_{R}(x_{0})$,$\frac{\partial v_{d}^{*}}{\partial n}=0$,$x\in\partial B_{R}(x_{0})$.

Proof of Lemma 3: Note$b(x)\leq\overline{b}(|x-x_{0}|)$,$x\in B_{R}(x_{0})$.

Thus, using $(v_{d}^{*})’(r)<0$ with $r=|x-x_{0}|$ , we have$-d\triangle v_{d}^{*}-f(x, v_{d}^{*})\geq-d^{2}v’’(r)-d^{2}\overline{b}v^{2}+(v_{d}^{*})^{3}\geq d^{2}(-v’’(r)-\overline{b}v^{2})=0$

.

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Let

be a minimizer of$\overline{v}_{d}$$\inf\{I(u;\Omega\backslash B_{R}(x_{0}))|u\in H^{1}(\Omega\backslash B_{R}(x_{0})),$

DefineNote
and thaton$\partial B_{R}(x_{0})\}$, andfor $x\in B_{R}(x_{0})$ .
forfunction. We want to show
and a piecewise. The following is a key lemma.
$x\in\Omega\backslash B_{R}(x_{0})$$\overline{u}_{d}(x)=\overline{v}_{d}(x)$$\overline{u}_{d}\in C(\overline{\Omega})$$u=v_{d}^{*}$$=v_{d}^{*}$$C^{1}$$\overline{u}_{d}(x)\geq\underline{u}_{d}(x),$.

$\overline{u}_{d}$is a supersolution$x\in\Omega$

Lemma 4 We have$\frac{\theta\overline{v}_{d}}{\partial n}\geq 0,$

where

is an inward unit normal on$n$$x\in\partial B_{R}(x_{0})$$\partial B_{R}(x_{0})$,.

Once we have this lemma, it is easy to see
Lemma 5$\overline{u}_{d}$is a supersolution.

Furthermore, the following lemma can be obtained by using Lemma 1.

Lemma 6$\overline{u}_{d}(x)\geq\underline{u}_{d}(x),$$x\in\Omega$.

Now, we have the following by using the argument of Brezis-Nirenberg( [2]), see alsoe.g. [12], [10] for Neumann boundary condition).

Theorem 5 Suppose the assumption $(A)$ . Then there enists a solution$x\in\Omega$$\overline{u}_{d}(x)\geq\tilde{u}_{d}(x)\geq\underline{u}_{4}(x),$Actually,$\tilde{u}_{d}$is a local minimizer of $I(u)$ on$H^{1}(\Omega)$$\tilde{u}_{d}$such that.

.

Actually,
tends to 1 uniformly on any compact subset of.
uniformly on any compact subset of
Finally, we give the outline of the proof of Lemma 4.
$\tilde{u}_{d}$$\Omega_{+}\backslash A$

and tends to$0$$\Omega_{-}\cup B_{R}(x_{0})$Outline of the proof of Lemma 4:

4.4

We claim the following. Claim 1:$\overline{v}_{d}(x)\leq dv(R)\equiv\alpha_{d}$,.

$x\in B_{R_{1}/4}(x_{0})\backslash B_{R}(x_{0})$. So, if this claim is true, Lemma 4 follows easily.
Noteon
To show Claim 1, we have two steps.
Step 1: We show,.
$\overline{v}_{d}=\alpha_{d}$$\partial B_{R}(x_{0})$$\overline{v}_{d}(x)\leq dv(R)$Proof of Step 1: FixLet$w_{d}$

and we want to show
, since $R_{1}>8R$ .

$x_{1}\in\partial B_{R_{1}/4}(x_{0})$$B_{R_{1}/8}(x_{1})\subset B_{R_{1}}(x_{0})\backslash B_{R}(x_{0})\subset\Omega_{-}$$x\in\partial B_{R_{1}/4}(x_{0})$$\overline{v}_{d}(x_{1})\leq\alpha_{d}$be a minimizer of$\inf\{I(u;B_{R_{1}/8}(x_{1}))|u-1\in H_{0}^{1}(B_{R_{1}/8}(x_{1}))\}$.

. Note that

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Since

$\overline{v}_{d}(x)\leq 1=w_{d}(x),$ $x\in\partial B_{R_{1}/8}(x_{1})$we have$\overline{v}_{d}(x)\leq w_{d}(x),$Lemma 1.
Finally, we show Claim 2: There exists a positive constantsuch that$(\#)$

If$(\#)$$0 \leq w_{d}(x_{1})\leq\frac{C_{0}d}{R_{1}^{2}}$$C_{0}$$x\in B_{R_{1}/8}(x_{1})$independent ofby$d,$ $R_{1}$.

is true,$\overline{v}_{d}(x_{1})\leq C_{0}\frac{d}{R_{1}^{2}}\leq\alpha_{d}=dv(R)$holds under the assumption$\frac{C_{0}}{R_{1}^{2}}\leq v(R)$.

Now, we can show the estimate in Claim 2 by comparing with the unique positive
solution of$\triangle U=U^{2},x\in B_{1}(0)$ ,$U(x)arrow+\infty(|x|arrow 1)$ .
For this boundary blow-up problem, see e.g., [4]. Next, we can show the following.
Step 2: Then, since, we can show$\overline{v}_{d}=dv(R)$$\overline{v}_{d}(x)\leq dv(R)$,$x\in B_{R_{1}/4}(x_{0})\backslash B_{R}(x_{0})$by using Lemma 1.

4.5

Keypoints of the proof for Theorem 4.

Concerning the proof of Theorem 4, we just remark the following keypoints. First ofall, we use a variational method and a sub-supersolution method as in the proof of
Theorem 2. In the construction and estimates of a suitable sub-supersolution, we useboundary blow-up solutions to the following problem:$-\triangle V=V^{2}-V^{3},$$V(x)=0(x\in\Gamma_{1})$,$V(x)>0$ ,

in

$G$

,

$V(x)=+\infty(x\in\Gamma_{2})$,where is an annular domain and. We note that existence of boundary
blow-up solutions to the equation above also seems new. For the details, see [8] (seealso [13] .
$G$$\partial G=\Gamma_{1}\cup\Gamma_{2}$$)$5Summary and Future Problems

Summary: We obtained non-trivial multiple stable patterns for reaction-diffusion
equation with (weak) Allee effect and for a balanced bistable reaction-diffusion equations under certain heterogeneous environments. The method are based on the construction of suitable super-sub solutions by using variational methods. To estimate
suitable sub-supersolution, we use certain boundary blow-up solutions.
Future Problems;$\bullet$.

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1. We imposed certain restriction to the configuration of the heterogeneous environments. Can we obtain under more general heterogeneous configuration?

2. Can we obtain the same results for environments which change smoothly?
3. How about for a reaction-diffusin system?

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