Fractional semilinear damped wave equation on the Heisenberg group

Abstract

This paper aims to investigate the Cauchy problem for the semilinear damped wave equation for the fractional sub-Laplacian $(-{\mathcal{L}}_{\mathbb{H}}{)}^{\alpha }$, $\alpha >0$ on the Heisenberg group ${\mathbb{H}}^n$ with power type non-linearity. With the presence of a positive damping term and nonnegative mass term, we derive $L^2-L^2$ decay estimates for the solution of the homogeneous linear fractional damped wave equation on ${\mathbb{H}}^n$, for its time derivative, and for its space derivatives. We also discuss how these estimates can be improved when we consider additional $L^1$-regularity for the Cauchy data in the absence of the mass term. Also, in the absence of mass term, we prove the global well-posedness for $2\le p\le 1+\frac{2\alpha }{(\mathcal{Q}-2\alpha {)}_{+}}$ $(\text{or }1+\frac{4\alpha }{\mathcal{Q}}<p\le 1+\frac{2\alpha }{(\mathcal{Q}-2\alpha {)}_{+}})$ in the case of $L^1\cap L^2$ $(\text{or }{L}^2)$ Cauchy data, respectively. However, in the presence of the mass term, the global (in time) well-posedness for small data holds for $1<p\le 1+\frac{2\alpha }{(\mathcal{Q}-2\alpha {)}_{+}}$. Finally, as an application of the linear decay estimates, we investigate well-posedness for the Cauchy problem for a weakly coupled system with two semilinear fractional damped wave equations with positive mass term on ${\mathbb{H}}^n$.