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

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 $p_{\mathrm{Fuj}}(\mathcal{Q})=1+2/\mathcal{Q}$, where $\mathcal{Q}$ is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for $p>p_{\mathrm{Fuj}}(\mathcal{Q})$ in an exponential weighted energy space. On the other hand, a blow-up result for $1<p\le p_{\mathrm{Fuj}}(\mathcal{Q})$ under certain integral sign assumptions for the Cauchy data by using the test function method.

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Cited by in corpus (6)

• Decay estimates for the linear damped wave equation on the Heisenberg group
• On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups
• On the critical exponent and sharp lifespan estimates for semilinear damped wave equations with data from Sobolev spaces of negative order
• A global existence result for a semilinear wave equation with lower order terms on compact Lie groups
• Liouville theorems for Kirchhoff-type hypoelliptic Partial Differential Equations and systems. I. Heisenberg group
• Blow-up solutions of damped Klein-Gordon equation on the Heisenberg group