Fujita exponent for heat equation with H\"{o}rmander vector fields

TL;DR

This paper investigates the critical Fujita exponent for the heat equation involving Hörmander vector fields, establishing conditions for global existence and blow-up, and analyzing nonlinearities with time-dependent factors.

Contribution

It extends the understanding of the Fujita exponent to heat equations with Hörmander vector fields and nonlinearities, providing new criteria for solution behavior.

Findings

Calculated the critical Fujita exponent for the heat equation with Hörmander vector fields.

Established necessary conditions for blow-up of solutions.

Provided sufficient conditions for global existence with time-dependent nonlinearities.

Abstract

In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on $\Rn$ satisfying H\"{o}rmander's rank condition with a non-linearity of the form $f(u)$, where $f$ is a suitable function and $u$ is the solution. In particular, when $f(u)=u^p$, we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type $\varphi(t)f(u)$.