Capacitary estimates of solutions of semilinear parabolic equations

TL;DR

This paper establishes capacitary estimates for positive solutions of a semilinear parabolic PDE, characterizing initial traces and blow-up sets using Bessel capacity, and proves uniqueness of solutions with given initial trace.

Contribution

It introduces a method to estimate solutions using Bessel capacity and characterizes initial traces and blow-up behavior, providing new insights into the solution structure.

Findings

Solutions are bounded by series involving Bessel capacity.

Existence and uniqueness of solutions with prescribed initial trace are proven.

Blow-up sets are characterized via the density of initial trace sets in terms of capacity.

Abstract

We prove that any positive solution of $ \prt_tu-\Delta u+u^q=0$ ($q>1$) in $\BBR^N\ti(0,\infty)$ with initial trace $(F,0)$, where $F$ is a closed subset of $\BBR^N$ can be estimated from above and below and up to two universal multiplicative constants, by a series involving the Bessel capacity $C_{2/q,q'}$. As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of $u(x,t)$ when $t\downarrow 0$, by using the "density" of $F$ expressed in terms of the $C_{2/q,q'}$-capacity.