# Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group

## TL;DR

This paper investigates the lifespan of solutions to the semilinear heat equation on the Heisenberg group, establishing critical exponents and sharp lifespan estimates using advanced test function techniques.

## Contribution

It identifies the Fujita exponent as critical for the Heisenberg group and provides sharp lifespan bounds for solutions in subcritical and critical cases.

## Key findings

- Fujita exponent $1 + 2/Q$ is critical for solution behavior.
- Sharp lifespan estimates are derived for different cases.
- Employs a revisited test function method for upper bounds.

## Abstract

In this paper we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group $\mathbf{H}_n$. The heat operator is given in this case by $\partial_t-\Delta_H$, where $\Delta_H$ is the so-called sub-Laplacian on $\mathbf{H}_n$. We prove that the Fujita exponent $1 + 2/Q$ is critical, where $Q = 2n + 2$ is the homogeneous dimension of $\mathbf{H}_n$. Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical case) we employ a revisited test function method developed recently by Ikeda-Sobajima. On the other hand, to find the lower bound estimate for the lifespan we prove a local in time result in weighted $L^\infty$ space.