Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space
by Yoshihisa Morita   Hirokazu Ninomiya  

Vol. 3 No. 4 (2008) P.567~P.584

ABSTRACT

We are dealing with a reaction-diffusion equation $u_t=\Delta u+u_{yy}+f(u)$ in $\mathbb{R}^{n+1}$, where $(x,y)=(x_1,\ldots,x_n,y)\in\mathbb{R}^{n+1}$ and $\Delta$ is the Laplacian in $\mathbb{R}^n$. Suppose that the equation has a bistable nonlinearity, namely it has two stable constant solutions $u=0,1$ and an unstable one between those. With the unbalanced condition $\int_0^1f(u)du>0$ the equation allows planar traveling waves connecting two constant solutions and an unstable standing solution $v(x)>0$ of $\Delta v+f(v)=0$ with $\lim_{|x|\to\infty}v(x)=0$. Then we show that there are a family of traveling waves $u=U(x,z),~z=y-ct$ connecting $u=1$ (or $u=0$) at $z=-\infty$ to $u=v(x)$ at $z=\infty$ with speeds belonging to a half infinite interval. The proof is carried out by using the comparison principle and constructing an appropriate subsolution and a supersolution. The existence theorem can be extended to a more general reaction-diffusion equation.

KEYWORDS
Reaction-diffusion equation, traveling wave, bistable nonlinearity

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35K57, 35B05, 35B40

MILESTONES

Received: 2008-08-12
Revised :
Accepted: 2008-08-12

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