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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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