Time-dependent singularities in semilinear parabolic equations: Behavior at the singularities

Abstract

Singularities of solutions of semilinear parabolic equations are discussed. A typical equation is ∂tu−Δu=up, x∈RN∖{ξ(t)}, t∈I. Here N≥2, p>1, I⊂R is an open interval and ξ∈Cα(I;RN) with α>1/2. For this equation it is shown that every nonnegative solution u satisfies ∂tu−Δu=up+Λ in D′(RN×I) for some measure Λ whose support is contained in {(ξ(t),t);t∈I}. Moreover, if (N−2)p<N, then u(x,t)=(a(t)+o(1))Ψ(x−ξ(t)) for almost every t∈I as x→ξ(t), where Ψ is the fundamental solution of Laplace's equation in RN and a is some function determined by Λ.