数理解析研究所講究録
第 1881 巻 2014 年 134-138

134

Diffusion-induced bifurcations from the stationarysolutions and infinity
総合数理学部 二宮広和 (Hirokazu Ninomiya) *School of Interdisciplinary Mathematical Sciences
Meiji University明治大学

Since A. Turing discovered the mechanism of periodic pattern generation induced
by diffusion in 1952, various patterns in nature have been explained by his idea, whichis called the diffusion-induced instab 漉 or I-劾短 $ngs$ instab 漉 $ty[8]$ . To explain this indetail, let us consider a system of the ordinary differential equations$ty$$[Matrix]$(1)

and the corresponding reaction-diffusion system$[Matrix]$(2)

in a bounded domain and $t>0$ with homogeneous Neumann boundary condition. We
note that any homogeneous solutions of (2) can be regarded as those of (1). Suppose that$(0,0)$ is a stable equilibrium of (1). Then $(0,0)$ is also a stationary solution of (2). However, $(0,0)$ may not be stable. If the nonlinear terms
and the diffusion coefficientsare chosen appropriately, then it becomes unstable. The stability is transferred toa spatially inhomogeneous steady state which bifurcates from the homogeneous one.
However, the diffusion-induced instability is caused only by a linear effect from themathematical viewpoint. We would like to extend this result to the changes of globaldynamics. It is also known that diffusion can be influenced on the global existence ofsolutions. More precisely, we are concerned with finite-time blow-up of solutions causedby diffusion. We say that a solution of (2) blows up in finite time if there is $T\in(0, \infty)$
such that$\Omega$$f_{1},$$d_{1},$$f_{2}$$d_{2}$$\lim_{t\nearrow}\sup_{T}(\Vert u_{1}(\cdot, t)\Vert_{L(\Omega)}\infty+\Vert v(\cdot, t)\Vert_{L^{\infty}(\Omega)})=\infty.$*This work is mainly based on the joint work [1]. The author is partially supported by a Research
Project Grant (A) from the Institute of Science and Technology, Meiji University, and the Challenging
Exploratory Research (No. 25610036) from the Japan Society for the PrQmotion of Science.

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Blow-up of solutions has been studied extensively by many authors. Most of them study
that a solution of (1) blow up in a finite time. Therefore the following question naturally
arises: can blow-up occur in (2), while (1) possesses a compact global attractor? Mizoguchi, Ninomiya and Yanagida [4] presented the affirmative answer. They constructed
the following system:(3)$\{\begin{array}{l}u_{t}=d_{1}\Delta u+|u-v|^{p-1}(u-v)-u,v_{t}=d_{2}\triangle v+|u-v|^{p-1}(u-v)-v\end{array}$with $p>1$ . and have shown that some solutions of (3) blow up in finite time if $0<d_{1}<$, while all solutions of the corresponding ordinary differential equations (1) converge
to $(0,0)$ as tends to infinity. This phenomenon is called diffusion-induced blowup. We
note that the diffusion-inhibited blowup is also studied in [3, 2].
Let us consider the change of the global dynamics more precisely. This result holds
even for large diffusivity and says that a compact attractor for (1) imbedded in (3) isonly local for (3). We notice that all solutions of (3) with $d_{1}=d_{2}>0$ are bounded for
all $t>0$ . We may expect that solutions bifurcate from the infinity whenvaries from$d_{2}$$t$$d_{1}$$d_{2}.$Along this context we refer to the bifurcation from infinity by Stuart [7] and Rabinowitz [6]. To explain their idea, let us consider the following nonlinear problem:(4)$u_{t}=\triangle u+au+f(u)$with Dirichlet homogeneous boundary condition. Suppose thatof- $A$ with Dirichlet boundary condition and$\lambda_{1,D}$

is the first eigenvalue(5)$f(u)=o(|u|)$astends to . The typical example of nonlinear term is $u/(1+u^{2})$ . Then the.
system possesses an unbounded bifurcation branch which includes.
We cannot apply this result to the system (3) because the condition (5) is not satisfied
for (3). We need to construct the theory for the bifurcation from infinity for the system.
The partial result will be presented in this short note. Seefor the details.
To begin with, we restrict the nonlinear terms
andto -homogeneous polynomials where is a positive integer. Namely,
$|u|$

$f$

$\infty$$(\lambda_{1,D}, \infty)$$\rfloor 1]$$\tilde{f}_{1}$$f_{2}$$p$$p$$\tilde{f}_{1}(su, sv)=s^{p}\tilde{f}_{1}(u, v) , \tilde{f}_{2}(su, sv)=s^{p}\tilde{f}_{2}(u, v)$

for $s>0.$
Consider$\{\begin{array}{ll}u_{t}=d_{1}u_{xx}+f_{1}(u, v) , x\in(0,1), t>0,v_{t}=d_{2}v_{xx}+f_{2}(u, v) , x\in(O, 1), t>0,u_{x}(0, t)=v_{x}(0, t)=u_{x}(1, t)=v_{x}(1, t)=0, t>0,\end{array}$

(6)

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where
$f_{1}(u, v)=\tilde{f}_{1}(u, v)-\epsilon u, f_{2}(u, v)=\tilde{f}_{2}(u, v)-\epsilon v.$

Assume that all solutions of the ordinary differential equations(7)$\{\begin{array}{l}U_{t} = f_{1}(U, V) ,V_{t} = f_{2}(U, V)\end{array}$exists globally in time and that (7) withpossesses an stationary solution
where. The homogeniety of nonlinearities implies$\epsilon=0$$(u_{*}, v_{*})$$u_{*}v_{*}\neq 0$$\{s(1, k_{*})|s\in \mathbb{R}, f(1, k_{*})=g(1, k_{*})=0\}$

is a set of stationary solutions of (7) withfollowing theorem.
Theorem 1 Consider the caseterms $f_{1}(u, v),$ $f_{2}(u, v)$ of (6) satisfy$\epsilon=0$(i) $f_{1}(1, k_{*})=f_{2}(1, k_{*})=0,$
(ii)
(iii)

$\epsilon=0$. Under these assumptions we have thein (6). Set $d_{2}=d_{1}\mu_{*}-\delta$ . Assume nonlinear$(k_{*}\neq 0, \infty)$,$\mu_{*}\neq\nu_{*},$$p$

is odd,(iv) $f_{1}(1, k_{*}+k_{*}\delta/d_{1}(\nu_{*}-\mu_{*}))>0$ for any$\delta$in some inetrval$\Lambda$

where$\mu_{*}:=\lim_{karrow k_{*}}\frac{f_{2}(1,k)}{kf_{1}(1,k)}, \nu_{*}:=\lim_{karrow k_{*}}\frac{d}{dk}\frac{f_{2}(1,k)}{f_{1}(1,k)}.$Then there exist an small intervalsuch thatof (6) for any

and a non-constant solution
and$\Lambda^{*}\subset\Lambda$$0\in\overline{\Lambda}^{*}$$\delta\in\Lambda^{*}$$\overline{k}=k_{*}+\frac{k}{d_{1}(\nu_{*}-\mu_{*})}*\delta+O(\delta^{2})$$(u_{*}, v_{*})=(\overline{u},\overline{k}\overline{u})$,$\overline{u}=\rho(\delta)U(x)^{1}$$\rho(\delta)=(\frac{d_{1}}{f_{1}(1,\overline{k})})^{1/(p-1)}$

where $U(x)$ is a solution of$\{\begin{array}{ll}U_{xx}+U^{p}=0, (0\leq x\leq L) ,U_{x}(0)=U_{x}(L)=0.

\end{array}$Moreover,$\lim_{\deltaarrow 0,\delta\in\Lambda^{*}}\overline{u}(x)=\infty, \lim_{\deltaarrow 0,\delta\in\Lambda^{*}}\overline{k}=k_{*}$.

(8)

137It follows from (8) that non-constant solutions bifurcates from the infinity.

Theorem 2 Assume $(i)-(iv)$ in Theorem 1 replaced with(ii)’ $\nu_{*}<\mu_{*}=1.$

Take
as in Theorem 1 and choose
satisfying $|(d_{2}-d_{1})/(d_{1}d_{2})|\leq\delta$ . Then there
is a positive constantsuch that a non-constant stationary solution of (6) exists forany.
$\Lambda^{*}$$d_{1},$$d_{2}$$\epsilon^{*}$$\epsilon\in(0, \epsilon^{*})$

Under the assumptions of Theorem 2, the non-constant solution in (6) bifurcatesfrom infinity as and go to due to Theorem 1.
We can confirm that$\delta$$\epsilon$$0$$\mu_{*}=1, \nu_{*}=0$

for the case (3). Thus Theorems 1 and 2 imply the existence of solutions with large
norm for (3). We can understand the diffusion-induced blowup in the following way.
The global attractor when $d_{1}=d_{2}$ is compact. However, when
, these solutions
come out from the infinity and these solutions and their unstable manifolds separate thebasin of domain. Thus some solutions which can go to infinity apper.
$d_{1}\neq d_{2}$oo$\infty$f$)$$\zeta J$$(a)d_{1}=d_{2}>0$

( )
$b$

$0<d_{1}<d_{2}$Figure 1: Heuristic understanding of the diffusion-induced blowup by the bifurcationfrom infinity. (a) all solutions converge to $0;(b)$ some large solution blows up in a finitetime.

These theorems facilitate to construct the candidate systems for the diffusion-inducedblowup. We only need to construct examples satisfying the assumptions of Theorem 2.
For example, we have$\{\begin{array}{ll}u_{t}=d_{1}u_{xx}+(u^{2}-v^{2})v-\epsilon u, x\in(0,1), t>0,v_{t}=d_{2}v_{xx}+(u^{2}-v^{2})u-\epsilon v, x\in(O, 1), t>0,u_{x}(0, t)=v_{x}(0, t)=u_{x}(1, t)=v_{x}(1, t)=0, t>0.\end{array}$

(9)

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Since $\mu_{*}=1,$ $v_{*}=-1$ , Theorem 2 immediately implies that the non-constant stationary
solutions bifurcate from the infinity.

References[1] C. Aida, C.-N. Chen and H. Ninomiya, in preparation.

[2] M. Fila and H. Ninomiya: Reaction versus diffusion: Blow-up induced and inhibitedby diffusivity, Russian Mathematical Surveys, 60 No. 6 (2005) 1217-1235[3] M. Fila, H. Ninomiya and J. L. Vazquez: Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems, Discrete and continuous
dynamical systems, 14 (2006), no. 1, 63-74.

[4] M. Mizoguchi, H. Ninomiya, and E. Yanagida: Diffusion-induced blowup in a nonlinear parabolic system, J. Dynamics and Differential Equations, 10-4 (1998), 619-638[5] H. Ninomiya and H. F. Weinberger: On p–homogeneous systems of differential equations and their linear perturbations, Applicable Analysis, 85 (2006), 225-246[6] P. H. Rabinowitz: On bifurcation from infinity, Journal of Differential Equations, 14(1973) 462-475[7] C. A. Stuart: Solutions of large norm for non-linear Sturm-Liouville problems, Quarterly Journal of Mathematics, 24 (1973) 129-139[8] A. M. Turing: The chemical basis of morphogenesis, Phil.
B237 no. 641, (1952) 37-72.

$\mathcal{I}Vans.$

Roy. Soc. London,