Blow up of solutions of semilinear heat equations in general domains

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Consider the nonlinear heat equation vₜ -Δv= |v|ᵖ⁻¹ v in a bounded smooth domain Ω⊂ ⁿ with n>2 and Dirichlet boundary condition. Given uₚ a sign-changing stationary 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. 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. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.