On asymptotics of solutions for superdiffusion and subdiffusion equations with the Riemann-Liouville fractional derivative

In the present paper, we focus on the study of the asymptotic behaviors of solutions for the Cauchy problem of time-space fractional superdiffusion and subdiffusion equations with integral initial conditions, where the Riemann-Liouville derivative is used in the temporal direction and the integral fractional Laplacian is applied in the spatial variables. The fundamental solutions of the considered equations, which can be represented in terms of the Fox $H$-function, are constructed and investigated by using asymptotic expansions of the Fox $H$-function. Then, we obtain the asymptotic behaviors of solutions in the sense of $L^p({\mathbb{R}}^d)$ and $L^{p,\mathrm{\infty }}({\mathbb{R}}^d)$ norms, where Young's inequality for convolution plays a very important role. Finally, gradient estimates and large time behaviors of solutions are also provided. In particular, we derive the optimal $L^2$- decay estimate for the subdiffusion equation.