Some Blow-Up Problems for a Semilinear Parabolic Equation with a Potential

TL;DR

This paper derives blow-up rate estimates for solutions to a semilinear parabolic equation with potential and improves understanding of blow-up time and set asymptotics under weaker conditions.

Contribution

It provides new blow-up rate estimates and refines the asymptotic analysis of blow-up behavior for solutions with large initial data.

Findings

Established blow-up rate estimates for the equation.

Improved asymptotic descriptions of blow-up time and set.

Weaker conditions than previous studies for blow-up analysis.

Abstract

The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=\Delta u+V(x) |u|^{p-1}u$ in $\Omega \times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data $u(x,0)=M\vf (x)$ as $M$ goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.