Min-max formulas for the speeds of multidimensional progressive waves. (Formules min-max pour les vitesses d’ondes progressives multidimensionnelles.) (French. English summary) Zbl 0956.35041
Summary: This work deals with travelling-wave solutions of some reaction-diffusion equations in infinite cylinders \(\Sigma= \{(x_1, y)\in\mathbb{R}\times \omega\}\) whose section \(\omega\subset \mathbb{R}^{n-1}\) \((n\geq 2)\) is a smooth domain. These waves have a speed \(c\) and a profile \(u\) which are solutions of
\[
\Delta u- \beta(y, c)\partial_{x_1}u+ f(u)= 0\quad\text{in }\overline\Sigma.
\]
For some types of nonlinearities \(f\), the speed \(c\) exists and is unique. We give a min-max variational formula for this speed \(c\). For another type of function \(f\), there exists a minimal speed, for which we give a min-max formula. In particular, these formulas generalize to the multidimensional case some results of A. I. Volpert, Vit. A. Volpert and Vl. A. Volpert [Traveling wave solutions of parabolic systems. Translations of Mathematical Monographs 140 (1994)] for planar waves.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 49J35 | Existence of solutions for minimax problems |
| 35A15 | Variational methods applied to PDEs |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
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