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

TL;DR

This paper investigates the existence, nonexistence, and behavior of solutions to the heat equation with superlinear absorption, focusing on solutions with time-dependent singular sets of specific dimensions.

Contribution

It establishes conditions for existence and nonexistence of solutions with singular sets, and characterizes the solutions' behavior near these sets.

Findings

No solutions when p ≥ (n-m)/(n-m-2)

Existence of two types of solutions when 1 < p < (n-m)/(n-m-2)

Solutions are unique and their behavior near singular sets is specified

Abstract

We consider the heat equation with a superlinear absorption term $\partial_{t} u-\Delta u= -u^{p}$ in $\mathbb{R}^n$ and study the existence and nonexistence of nonnegative solutions with an $m$-dimensional time-dependent singular set, where $n-m\geq 3$. First, we prove that if $p\geq (n-m)/(n-m-2)$, then there is no singular solution. We next prove that, if $1<p<(n-m)/(n-m-2)$, then there are two types of singular solution. Moreover, we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set.