Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order

Abstract

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian \(\mathcal {L}\) on the Heisenberg group \(\mathbb {H}^n\) with power type nonlinearity \(|u|^p\) and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space \(\dot{H}^{-\gamma }_{\mathcal {L}}(\mathbb {H}^n), \gamma >0\), on \(\mathbb {H}^n\). In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent \(p_{\text {crit}}(Q, \gamma )=1+\frac{4}{Q+2\gamma },\) for \(\gamma \in (0, \frac{Q}{2})\), where \(Q:=2n+2\) is the homogeneous dimension of \(\mathbb {H}^n\). More precisely, we establish

• A global-in-time existence of small data Sobolev solutions of lower regularity for \(p>p_{\text {crit}}(Q, \gamma )\) in the energy evolution space;

• A finite time blow-up of weak solutions for \(1<p<p_{\text {crit}}(Q, \gamma )\) under certain conditions on the initial data by using the test function method.

Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.