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

Abstract

SynopsisWith 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 Δw + λf(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.