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

2013/04/09 by Dickstrein, Flavio, Pacella, Filomena, Scunzi, Berardino
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1304.2571

Abstract

Consider the nonlinear heat equation v_t-Δv=|v|^p-1v in the unit ball of R², 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 λu_p,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,K and of the linearized operator L= -Δ- p |u_p,K|^p-1. To show this we consider the linearized operator L= -Δ- p|u_p|^p-1 and study the behavior of its first eigenvalue and of its first normalized eigenfunction for large p.

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