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

抄録

This paper is concerned with positive solutions of the semilinear diffusion equation u_t=\ riangle u+u^p in Ω under the Neumann boundary condition, where p>1 is a constant and Ω is a bounded domain in \bm{R}^N with C² boundary. This equation has the constant solution (p-1)^-1/(p-1)(T₀-t)^-1/(p-1)(0≤ t<T₀) with the blow-up time T₀>0. It is shown that for any ε>0 and open cone Γ in {f∈ C(overline{Ω})|f(x)>0}, there exists a positive function u₀(x) in overline{Ω} with ∂ u₀/∂ v=0 on ∂Ω and \left//u₀(x)-(p-1)^-1/(p-1)T₀^-1/(p-1)\ ight//_{C²(overline{Ω})}|<ε 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.