Blow up of solutions of semilinear heat equations in general domains

Consider the nonlinear heat equation vt - Delta v = |v| (p-1)v in a bounded smooth domain Omega subset of 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 v(up) blows up in finite time if |v -1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent n+2/n-2. Since for v = 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.