Variable instability of a constant blow-up solution in a nonlinear heat equation

Abstract

This paper is concerned with positive solutions of the semilinear diffusion equation ut=∆u+up in Ω under the Neumann boundary condition, where p>1 is a constant and Ω is a bounded domain in RN with C2 boundary. This equation has the constant solution (p-1)-1/(p-1)(T0-t)-1/(p-1)(0≤t 0. It is shown that for any e>0 and open cone Γ in {f∊C(Ω¯)|f(x)>0}, there exists a positive function u0(x) in Ω¯ with ∂u0/∂v=0 on ∂Ω and ||u0(x)-(p-1)-1/(p-1)T0-1/(p-1)||C2(Ω¯)<e such that the blow-up time of the solution u(x,t) with initial data u(x,0)=u0(x) is larger than T0 and the function u(x,T0) belongs to the cone Γ. A theorem on the blow-up profile is also given.