Blow up of solutions of semilinear heat equations in general domains

Consider the nonlinear heat equation vt − Δv = |v|p−1v in a bounded smooth domain Ω ⊂ Rn with n > 2 and Dirichlet boundary condition. Given up a sign-changing stationary classical solution fulfilling suitable assumptions, we prove that the solution with initial value ϑup blows up in finite time if |ϑ − 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent n+2/n−2. 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.