In this article, we focus on the blow-up problem of solution to Caputo-Hadamard fractional diffusion equation with fractional Laplacian and nonlinear memory. By virtue of the fundamental solutions of the corresponding linear and nonhomogeneous equation, we introduce a mild solution of the given equation and prove the existence and uniqueness of local solution. Next, the concept of a weak solution is presented by the test function and the mild solution is demonstrated to be a weak solution. Finally, based on the contraction mapping principle, the finite time blow-up and global solution for the considered equation are shown and the Fujita critical exponent is determined. The finite time blow-up of solution is also confirmed by the results of numerical experiment.

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.