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

Singularities of solutions of semilinear parabolic equations are discussed. A typical equation is ∂tu−Δu=up∂tu−Δu=up, x∈RN∖{ξ(t)}x∈RN∖{ξ(t)}, t∈It∈I. Here N≥2N≥2, p>1p>1, I⊂RI⊂R is an open interval and ξ∈Cα(I;RN)ξ∈Cα(I;RN) with α>1/2α>1/2. For this equation it is shown that every nonnegative solution u   satisfies ∂tu−Δu=up+Λ∂tu−Δu=up+Λ in D′(RN×I)D′(RN×I) for some measure Λ whose support is contained in {(ξ(t),t);t∈I}{(ξ(t),t);t∈I}. Moreover, if (N−2)p