Blow up of solutions of semilinear heat equations in non radial domains of $mathbb{R}^2$

We consider the semilinear heat equation (P_p) -v_t - Δv = |v|^{p-1}v in Ω x (0, T); v = 0 on δΩ × (0, T); v(0) = v_0 in Ω; where p ≥ 1, Ω is a smooth bounded domain of ℝ^2, T ∈ (0;+∞] and v_0 belongs to a suitable space. We give general conditions for a family up of sign-changing stationary solutions of (P_p), under which the solution of (P_p) with initial value v_0 = λ u_p blows up in finite time if |λ - 1| > 0 is sufficiently small and p is sufficiently large. Since for λ = 1 the solution is global, this shows that, in general, the set of the initial conditions for which the solution is global is not star-shaped with respect to the origin. In [Dickstein,Pacella,Sciunzi, Journal of Evolution Equation, 2014] this phenomenon has been previously observed in the case when the domain is a ball and the sign changing stationary solution is radially symmetric. Our conditions are more general and we provide examples of stationary solutions up which are not radial and exhibit the same behavior.