The Fujita exponent for a semilinear heat equation with forcing term on Heisenberg Group

TL;DR

This paper investigates the critical exponent for a semilinear heat equation with forcing on the Heisenberg group, revealing boundedness across all dimensions and providing lifespan estimates, contrasting with Euclidean cases.

Contribution

It introduces a novel method using nonlinear capacity estimates tailored to the Heisenberg group to determine the critical exponent.

Findings

Critical exponent is bounded for all Heisenberg group dimensions.

Contrasts with Euclidean case where the exponent is infinite in 1D and 2D.

Provides an upper bound on the lifespan of solutions.

Abstract

In this paper, we study a critical exponent to the semilinear heat equation with forcing term on Heisenberg group. Our technique of proof is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg group. Surprisingly, the critical exponent will be bounded for all dimensions of the Heisenberg group, in contrast to the Euclidean case which in the 1D and 2D cases will be infinite. In addition, we give an upper bound estimate on the lifespan of local solutions.