A reaction-diffusion system approximation (Free Boundary Problems)
Hirokazu Ninomiya · Institutional Repositories DataBase (IRDB) · 2001
The theory of reaction-diffusion systems has been developed in the last two decades.Especially, asingular limit analysis reveals various behavior of solutions to us.It has mainly been used for purpose of studying the dynamics of the specific $\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}-\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ system.In this paper, the following type of equations is called areaction-diffusion system: $u_{t}=D\triangle u+f(u)$ , (1.1) in $x\in\Omega\subset \mathrm{R}^{N}$ , $t>0$ with the Neumann homogeneous boundary condition and initial condition where $u={}^{t}(u_{1}, \cdots, u_{M})$ , $f(u)={}^{t}(f_{1}(u), \cdots, f_{M}(u))$ , and $D$ is adiagonal matrix whose elements are positive (or non-negative).Areaction- diffusion system consists of two parts: one is kinetic term $f$ ;the other is adiffusion one $D\triangle$ .One might think that the diffusion term makes the solution spatially homogenize.After discovering Turing's instability, it turns out that one's intuition might not apply all the reaction-diffusion system (cf.[14, 8, 12]).We encounter the questions: How wide is aclass of reaction-diffusion systems?, or How rich are the dynamics of reaction-diffusion systems?Thus we study the relation- ship between reaction-diffusion systems and the following two systems:(i) the tw0-phase Stefan problem, (ii) the cross-diffusion system.Obviously, these systems do not belong to aclass of reaction-diffusion systems.In this paper, new types of reaction-diffusion systems with asmall parameter are proposed, which converge to the above two system (i) and (ii) respectively.That is, by the singular limit analysis, it is shown that any solution of these systems converges to that of (i) or (ii) respectively.In other word, the above two system (i) and (ii) can be embedded in the class of reaction-diffusion systems.