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Travelling fronts in cylinders. (English) Zbl 0799.35073
The authors are concerned with travelling wave solutions in an infinite cylinder \(\Sigma:= \mathbb{R}\times \omega\) with \(\omega\subseteq \mathbb{R}^{n- 1}\) a bounded domain. They consider equations of the form \(\Delta u- \beta(y,c) \partial_{x_ 1} u+ f(u)=0\) in \(\Sigma\) \((x=(x_ 1,y))\) under homogeneous Neumann boundary conditions on \(\partial\Sigma\) and asymptotic conditions \(u(-\infty, \cdot)=0\) and \(u(+\infty, \cdot)=1\). As far as \(\beta\) is concerned, they assume: \(\beta\) continuous on \(\omega\times \mathbb{R}\) and strictly increasing in its second argument, \(\beta(y,c)\to \pm\infty\) as \(c\to \pm\infty\) uniformly for \(y\in\omega\). One may think of \(f\) as being in \(C^ 2([ 0,1])\) with \(f(0)= 0= f(1)\) and \(f'(1)< 0\).
Three cases are considered: (A) \(f>0\) on \((0,1)\); (B) \(\exists\theta>0: f|_{[0,\theta]}\equiv 0\) and \(f|_{(\theta,1)} > 0\); (C) \(\exists\theta>0: f|_{(0,\theta)} < 0\) and \(f|_{(\theta,1)} > 0\). In case (A) they show that there exists a \(c^*\in\mathbb{R}\) such that the above problem is solvable, iff \(c\geq c^*\). If \(f'(0)>0\), then the solution is unique modulo translations. For case (B) they obtain a solution \((c,u)\), whereas \(\omega\) convex has to be additionally required in case (C) for that purpose.
There are many more significant results in this comprehensive investigation, which extends various classical results from combustion theory as well as the celebrated paper of Kolmogorov, Petrovsky and Piskounov to higher dimensions.
Reviewer: G.Hetzer (Auburn)

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35K99 Parabolic equations and parabolic systems
80A25 Combustion
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