# Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity

## TL;DR

This paper determines the critical exponent for the semilinear damped wave equation on the Heisenberg group, showing global existence for supercritical powers and blow-up for subcritical powers, thus extending Fujita's theory to this setting.

## Contribution

It establishes the critical Fujita exponent for the damped wave equation on the Heisenberg group and analyzes solution behavior relative to this exponent.

## Key findings

- Critical exponent is p_Fuj = 1 + 2 / Q.
- Global solutions exist for p > p_Fuj.
- Solutions blow up for 1 < p ≤ p_Fuj.

## Abstract

In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 / \mathscr{Q}$, where $\mathscr{Q}$ is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for $p >p_{\mathrm{Fuj}}(\mathscr{Q})$ in an exponential weighted energy space. On the other hand, a blow-up result for $1 < p \leq p_{\mathrm{Fuj}}(\mathscr{Q})$ under certain integral sign assumptions for the Cauchy data by using the test function method.