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Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity

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  • This paper is concerned with steady-state solutions for the following reaction-diffusion equation $$ u_t= \varepsilon^2 u_{xx} + u(1-u)(u-a(x)),\quad(x,t)\in (0,1)\times(0,\infty) $$ with $u_x(0,t) = u_x(1,t) = 0$ for $t\in (0,\infty)$. Here $\varepsilon$ is a small positive parameter and $a$ is a $C^2[0,1]$ function such that $0 < a(x) < 1$ for $x\in [0,1]$ and that $\Sigma:= \{x\in(0,1); a(x) = 1/2\}$ is a nonempty finite set. It is well known that the corresponding steady-state problem admits solutions with transition layers or spikes when $\varepsilon$ is sufficiently small. We will give some information on the location of transition layers and spikes for steady-state solutions. Under certain circumstances, such solutions possess multi-layers or multi-spikes. We will also show some conditions for the appearance of multi-spikes as well as for the existence of multi-layers.
    Mathematics Subject Classification: 35K57, 35B25.

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