Convergence of bounded solutions to gradient‐like semilinear Cauchy problems with radial nonlinearity

Article type: Research Article

Authors: Chill, Ralph

Affiliations: Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany E‐mail: chill@mathematik.uni‐ulm.de

Note: [] The author is supported by Deutscher Akademischer Austauschdienst (Stipendium im Rahmen des gemeinsamen Hochschulsonderprogramms III von Bund und Ländern über den DAAD). This support is gratefully acknowledged.

Abstract: We prove convergence to a steady state of bounded solutions of the abstract first order semilinear Cauchy problem ut+Lu+g(Ψ(u))Cu=0, t∈R+, and of the second order semilinear Cauchy problem utt+αut+Lu+g(Ψ(u))Cu=0, t∈R+. We apply the abstract results to semilinear parabolic and hyperbolic partial differential equations including the heat equation, the wave equation, a Kuramoto–Sivashinsky model and the Kirchhoff–Carrier equation.

Keywords: semilinear, gradient‐like, heat equation, wave equation, Kuramoto–Sivashinsky model, Kirchhoff–Carrier equation

Journal: Asymptotic Analysis, vol. 33, no. 2, pp. 93-106, 2003