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Travelling waves for reaction–diffusion equations with time depending nonlinearities. (English) Zbl 1032.35089
This paper concerns the parabolic equation $v_t =\Delta v-f(v,\alpha \cdot x-ct),\tag{1}$ where $$v=(v^1,\dots ,v^k):\mathbb{R}^l\times \mathbb{R}\to \mathbb{R}^k$$, $$x,\alpha\in \mathbb{R}^l$$, $$|\alpha|=1$$, $$t\in \mathbb{R}$$, $$c>0$$ and $$f:\mathbb{R}^k\times \mathbb{R}\to \mathbb{R}^k$$ is a continuous function. Under appropriate assumptions the author proves that equation (1) admits a travelling wave solution with end points $$w_{\pm}$$, i.e., a solution $$v$$ of the form $$v(x,t)=w(\alpha\cdot x-ct)$$ and such that $$\lim_{x\to\pm\infty}w(x)=w_{\pm}\in \mathbb{R}^k$$. The proof is obtained by reduction of the equation (1) to an ordinary one and by using the Schauder fixed point theorem and a suitable diagonalization procedure.

##### MSC:
 35K57 Reaction-diffusion equations 35K40 Second-order parabolic systems
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##### References:
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