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Concentration Phenomenon of Semiclassical States to Reaction-Diffusion System

Tianxiang GouZhitao Zhang
arXiv: Analysis of PDEs
Sep 2019
摘要
We consider the concentration phenomenon of semiclassical states to the following $2M$-component reaction-diffusion system on $\R \times \R^N$, \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 $(u, v): \R \times \R^N \to \R^M \times \R^M$. The system describing reaction and diffusion of chemicals or morphogens arises in theoretical physics, chemistry and biology. It is shown in this paper that ground states to the system concentrate around the local minimum points of $V$ as $\eps \to 0^+$. Our approach is variational, which is based upon a new linking-type argument as well as iteration techniques.
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