Excluding blowup at zero points of the potential by means of Liouville-type theorems

TL;DR

This paper establishes local Liouville-type theorems to exclude blowup at zero points of the potential in nonlinear heat equations, advancing understanding of solution behavior near potential blowup points.

Contribution

It provides a local version of a known global result and applies Liouville theorems to rule out blowup at zero potential points for certain nonlinear heat equations.

Findings

Blowup at zero points of the potential is excluded for monotone solutions.

The results hold in both Sobolev subcritical and radial cases.

The approach uses Liouville-type theorems to analyze solution behavior.

Abstract

We prove a local version of a (global) result of Merle and Zaag about ODE behavior of solutions near blowup points for subcritical nonlinear heat equations. As an application, for the equation $u_t= \Delta u+V(x)f(u)$, we rule out the possibility of blowup at zero points of the potential $V$ for monotone in time solutions when $f(u)\sim u^p$ for large $u$, both in the Sobolev subcritical case and in the radial case. This solves a problem left open in previous work on the subject. Suitable Liouville-type theorems play a crucial role in the proofs.