Time-dependent singularities in a semilinear parabolic equation with absorption

Abstract

This paper concerns solutions with time-dependent singularities for a semilinear parabolic equation with a superlinear absorption term. Here, by time-dependent singularity, we mean a singularity with respect to the space variable whose position depends on time. It is shown that if the power of the nonlinearity is in some range, then any singularity is removable. On the other hand, in other range, two types of time-dependent singular solutions exist: One resembles the fundamental solution of the Laplace equation near the singular point, and the other has a stronger singularity.

On the evolution equation with a dynamic Hardy-type potential

“…In this paper, we consider the case where both the position and strength of the singularity depend on time. See [7,14,20,22] for related works on such dynamic singularities.…”

Solutions with time-dependent singular sets for the heat equation with absorption

“…Let us return to the parabolic problem (1.1), which we treat in this paper. As parabolic analogs of the elliptic results introduced above, the first author and Yanagida [25] considered the case m = 0, that is, the case where the singular set is a time-dependent point M t = {ξ(t)}. For p < p sg , their results imply that there are singular solutions satisfying (A') and (B'), and that there is no singular solution satisfying (C') or (D'), where these conditions are parabolic analogs of (A), (B), (C) and (D) introduced later.…”

“…Recently, Hirata [14] employed the Minkowski content of M with respect to the parabolic distance, and gave removability results for each of the nonnegative solutions of (1.1). On the existence of singular solutions, more recently, the first author and Yanagida [25] studied singular solutions in the specific case where each M t is a time-dependent point. The novelty of their results is the existence of singular solutions for p < p sg := n n − 2 .…”

Solutions with snaking singularities for the fast diffusion equation

“…For various results on solutions with moving singularities (θ ≡ 0) for the heat equation (m = 1) we refer to [13,14,24], for semilinear heat equations see [10,11,14,15,[19][20][21][22][23]25] and also [16,17] for the Navier-Stokes system.…”