Concentration Phenomenon of Semiclassical States to Reaction-Diffusion Systems

TL;DR

This paper investigates how semiclassical states of a reaction-diffusion system concentrate around local minima of a potential function, using variational methods and novel analytical techniques.

Contribution

It introduces a new linking-type variational approach to prove the existence of concentrating semiclassical states in reaction-diffusion systems.

Findings

Semiclassical states concentrate near local minima of V.

Existence of solutions is established under mild conditions.

The approach combines variational methods with iterative and interior estimates.

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.