New definitions of decay indicators and critical exponent for fractional semi-linear structurally damped evolution equations on the Heisenberg group

Abstract

In this paper, we introduce the lower and upper decay indicators and the associated decay character on the Heisenberg group. These notions are used to derive decay estimates for the linear fractional diffusion equation and to characterize the decay of initial data in several function spaces. We then study the Cauchy problem $$ \partial_t^2u+\left(-\Delta{\mathrm H}\right)^{\delta_1}u+\left(-\Delta_{\mathrm H}\right)^{\delta_2}\partial_tu=0, \quad \delta_1\in\left[0,\frac{\delta_2}{2}\right], $$ and establish decay estimates for solutions and their derivatives in homogeneous fractional Sobolev spaces in terms of the decay characters of the initial data. These results recover known estimates and extend them to new classes of data. We also investigate the corresponding semilinear problem with nonlinearity $|u|^p$. Global existence and decay are proved for $$ p>1+\frac{2\omega\delta_1}{Q-2\omega\delta_2}, \quad \omega=\frac{Q}{Q+\min{r_{\mathrm H}(u_0),r_{\mathrm H}(u_1)-2\delta_2}+2\delta_2}, $$ together with the corresponding critical case. Finally, by constructing test functions adapted to the nonlocal fractional sub-Laplacians, we establish blow-up results and identify the critical exponent $$ p=1+\frac{2m\delta_1}{Q+m\gamma-2m\delta_2} $$ for initial data in $\dot H_m^{-\gamma}(\mathbf H_n)$, where $m\in(1,2]$ and $\gamma\in\left[0,Q-\frac{Q}{m}\right)$.