Layers and spikes in non-homogeneous bistable reaction-diffusion equations

Ai, Shangbing; Chen, Xinfu; Hastings, Stuart

doi:10.1090/S0002-9947-06-03834-7

Layers and spikes in non-homogeneous bistable reaction-diffusion equations
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by Shangbing Ai, Xinfu Chen and Stuart P. Hastings PDF

Trans. Amer. Math. Soc. 358 (2006), 3169-3206 Request permission

Abstract:

We study $\varepsilon ^2\ddot {u}=f(u,x)=A u (1-u) (\phi -u)$, where $A=A(u,x)>0$, $\phi =\phi (x)\in (0,1)$, and $\varepsilon >0$ is sufficiently small, on an interval $[0,L]$ with boundary conditions $\dot {u}=0$ at $x=0,L$. All solutions with an $\varepsilon$ independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to $f=A(u,x)\; (u-\phi _-) (u-\phi ) (u-\phi _+)$ with $\phi _-(x)<\phi (x)<\phi _+(x)$ and also to an infinite interval.

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