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

We prove convergence to a steady state of bounded solutions of the abstract first order semilinear Cauchy problem u_t+Lu+g(Ψ(u))Cu=0, t∈R₊, and of the second order semilinear Cauchy problem u_tt+αu_t+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.