Global existence and convergence to a singular steady state for a semilinear heat equation

Abstract

Synopsis With certain initial and boundary conditions the solution u * to the semilinear heat equation ∂ u */∂ t = ∂ u * + λ * f ( u *), where f is a positive superlinear function and λ is the supremum of the open spectrum for the steady state problem Δ w + λ f (w) = 0, is found to exist for all time and to be unbounded. Moreover u * approaches w * a singular steady state, as / tends to infinity.