Nonexistence of solutions of certain semilinear heat equations

TL;DR

This paper investigates the existence and nonexistence of solutions for certain semilinear heat equations with space-dependent forcing, establishing bounds and applying the approach to various vector fields and differential operators.

Contribution

It introduces a method to prove nonexistence of global solutions and applies it to diverse vector fields, providing bounds on blow-up times and critical exponents.

Findings

Proved local existence of solutions using Banach fixed-point theorem.

Established nonexistence of global weak solutions with test functions.

Derived upper bounds for blow-up times in various cases.

Abstract

We consider a semilinear heat equation involving a forcing term which depends only on the space variable. To start with, the existence of a local mild solution is proved through an application of the Banach fixed-point theorem. With the help of carefully defined test functions, we then prove the nonexistence of global weak solutions. The most crucial step is to find the function $d(x)$ used in our proofs, which seems to depends only upon the considered vector fields. This leads to lower bounds for a possible critical Fujita-type exponent. The same function $d(x)$ could lead to a potential norm function which would be most suitable while working with these vector fields. Section 4 is the attraction of this paper in which we apply our approach to all of the vector fields discussed by Biagi, Bonfiglioli and Bramanti, giving rise to Grushin-type and Engel-type PDOs, and more. An upper bound for the blow-up time of local solutions is also provided in each of these cases.