Lyapunov stability of planar waves to the reaction-diffusion equation with a non-Lipschitzian reaction term
Summary: Extending [\textit{P. Drábek} and \textit{P. Takáč}, J. Math. Biol. 75, No. 4, 929--972 (2017; Zbl 1378.35311)], we investigate the Lyapunov stability of planar waves for the reaction-diffusion equation on \(\mathbb{R}^n\), \(n\geq2\), with a \(\alpha\)-Hölder continuous (\(0<\alpha<1\)), but not necessarily smooth reaction term. We first consider an initial value problem for the equation and then construct sub- and supersolutions to the problem by a subtle modification of the planar wave. Our main result states that a bounded classical solution to the problem stays near the planar wave for all time whenever an initial data is close enough to the planar wave.
scientific article; zbMATH DE number 7433508