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Travelling waves for semilinear parabolic partial differential equations in cylindrical domains. (English) Zbl 0699.35137
Heidelberg: Univ. Heidelberg, Naturwiss.-Math. Gesamtfakultät, Diss. 44 p. (1988).
The author investigates the problem $u_ t=\Delta_{\xi,y}u+f(u)\quad for\quad (t,\xi,y)\in {\mathbb{R}}^+\times {\mathbb{R}}\times \Omega,\quad u=0\quad for\quad (t,\xi,y)\in {\mathbb{R}}^+\times {\mathbb{R}}\times \partial \Omega$ with $$f(0)=f(1)=0$$, $$f'(1)<0$$, $$\Omega \subset {\mathbb{R}}^ n$$ a bounded domain and looks for solutions in the form of a travelling wave $$u(\xi +ct,y)=u^*(x,y).$$ Let $$\mu$$ denote the first eigenvalue of $$- \Delta_ y$$ with zero boundary condition on $$\partial \Omega$$. The travelling wave solution $$u^*$$ exists: (1) for a certain c if $$f'(0)<\mu$$ and $\inf_{v\in H^ 1_ 0(\Omega)}\int_{\Omega}((1/2)| \nabla v|^ 2- \int^{v}_{0}f(\tau) d\tau) dy<0;$ (2) for all c greater than a certain $$c^*$$ if $$f'(0)>\mu$$ and $$f(u)>0$$, $$u\in (0,1).$$
In both cases $$u^*(-\infty,y)=0$$ and $$u^*(\infty,y)=u^+(y)$$ where $$u^+$$ is a positive solution to $$\Delta_ yv-f(v)=0$$ in $$\Omega$$, $$u| \partial \Omega =0$$. A number of results on monotonicity, uniqueness and other properties is proved.
Reviewer: O.Vejvoda

MSC:
 35K55 Nonlinear parabolic equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35C05 Solutions to PDEs in closed form