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

Singularities of solutions of semilinear parabolic equations are discussed. A typical equation is partial derivative(t)u - Delta u = u(p), x is an element of R-N \ {xi(t}, t is an element of I. Here N >= 2, p > 1, I subset of R is an open interval and xi is an element of C-alpha (I; R-N) with alpha > 1/2. For this equation it is shown that every nonnegative solution u satisfies partial derivative(t)u - Delta u = u(p) + Lambda in D' (R-N x I) for some measure Lambda whose support is contained in {(xi(t), t); t is an element of I}. Moreover, if (N - 2)p < N, then u(x, t) = (a(t) + o(1))Psi (x - xi(t)) for almost every t is an element of I as x -> xi(t), where Psi is the fundamental solution of Laplace's equation in R-N and a is some function determined by Lambda. (C) 2016 Elsevier Inc. All rights reserved.