Travelling fronts in cylinders
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.
scientific article; zbMATH DE number 220980
scientific article