Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two

Consider the nonlinear heat equation $$v_t -Delta v=|v|^{p-1}v qquad qquad qquad (NLH)$$ in the unit ball of ({mathbb{R}^2}) , with Dirichlet boundary condition. Let ({u_{p,mathcal{K}}}) be a radially symmetric, sign-changing stationary solution having a fixed number ({mathcal{K}}) of nodal regions. We prove that the solution of (NLH) with initial value ({lambda u_{p,mathcal{K}}}) blows up in finite time if |λ ?1| > 0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of ({u_{p,mathcal{K}}}) and of the linearized operator ({L= -Delta - p | u_{p,mathcal{K}} | ^{p-1}}) .