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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

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