×

Travelling front solutions of semilinear equations in \(n\) dimensions. (English) Zbl 0780.35054

Frontiers in pure and applied mathematics, Coll. Pap. Ded. J.-L. Lions Occas. 60th Birthday, 31-41 (1991).
[For the entire collection see Zbl 0722.00015.]
An infinite cylindrical domain \(\Sigma=\mathbb{R}\times \omega\subset\mathbb{R}^ N\), where \(\omega\) is a bounded domain in \(\mathbb{R}^{N-1}\) with smooth boundary, is considered. An element \(x\in\Sigma\) is written in the form \(x=(x_ 1,y)\), \(x_ 1\in\mathbb{R}\), \(y=(x_ 2,\dots,x_ n)\in\omega\) and by \(\nu\) is denoted the outward unit normal vector on \(\partial\omega\) as well as the outward unit normal to \(\partial\Sigma\).
Travelling front solutions in \(\Sigma\) are solutions of problems of the following type: \[ -\Delta u+(c+\alpha(y))u_{x_ 1}=f(u) \qquad (\text{or } -\Delta u+c\alpha(y)u_{x_ 1}=f(u)) \text{ in } \Sigma, \] with \(\partial u/\partial\nu=0\) on \(\partial\Sigma\), \(u(-\infty,y)=0\), \(u(+\infty,y)=1\), uniformly in \(y\in\overline{\omega}\). Here \(\alpha: \overline{\omega}\to \mathbb{R}\) is a given continuous function assumed to be positive and \(c\) is a real parameter, the velocity, usually an unknown in the problem. The function \(f\) will be assumed to be Lipschitz, and to vanish outside the interval [0,1]; on the interval [0,1] it is assumed that \(f\in C^{1,\delta}\) for some \(0<\delta<1\) on some neighbourhood of 0 and 1, respectively, and \(f'(1)<0\). The existence and uniqueness theorems of \((c,u)\) and the exponential behaviour of \(u\) as \(x\to-\infty\) are presented.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K57 Reaction-diffusion equations

Citations:

Zbl 0722.00015
PDFBibTeX XMLCite