Blow up of solutions of semilinear heat equations in general domains

Source abstract

Consider the nonlinear heat equation v t - Δv = |v| p-1 v in a bounded smooth domain Ω ⊂ ℝ n with n > 2 and Dirichlet boundary condition. Given u p a sign-changing stationary classical solution fulfilling suitable assumptions, we prove that the solution with initial value ϑu p blows up in finite time if |ϑ - 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent [Formula: see text]. Since for ϑ = 1 the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped with respect to the origin. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.