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

Abstract

Consider the nonlinear heat equation v_t-\Delta v=|v|^{p-1}v in the unit ball of R^2, with Dirichlet boundary condition. Let u_{p,K} be a radially symmetric, sign-changing stationary solution having a fixed number K of nodal regions. We prove that the solution of the equation with initial value \lambda u_{p,K} blows up in finite time if |\lambda-1|>0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of $u_{p,K}$ and of the linearized operator L= -\Delta - p |u_{p,K}|^{p-1}. To show this we consider the linearized operator L= -\Delta - p|u_p|^{p-1} and study the behavior of its first eigenvalue and of its first normalized eigenfunction for large p.