Concentration Phenomenon of Semiclassical States to Reaction-Diffusion Systems

Abstract

In this paper, we consider concentration phenomenon of semiclassical states to the following $2M$-component reaction-diffusion system in $\R \times \R^N$, \begin{align*} \left\{ \begin{aligned} \partial_t u &=\eps^2 \Delta_x u-u-V(x)v + \partial_v H(u, v),\\ \partial_t v &=-\eps^2 \Delta_x v+v + V(x)u - \partial_u H(u, v), \end{aligned} \right. \end{align*} where $M \geq 1$, $N \geq 1$, $\eps>0$ is a small parameter, $V \in C^1(\R^N, \, \R)$, $H \in C^1(\R^M \times \R^M, \, \R)$ and $(u, v): \R \times \R^N \to \R^M \times \R^M$. It is proved that there exist semiclassical states concentrating around the local minimum points of $V$ under mild assumptions. The approach is variational, which is mainly based upon a new linking-type argument, iterative techniques and interior estimates for nonlinear parabolic equations.