Asymptotic behaviours of solution to Caputo–Hadamard fractional partial differential equation with fractional Laplacian

Abstract

In this paper, we study the asymptotic behaviours of solution to time–space fractional diffusion equation, where the time derivative with order α is in the sense of Caputo–Hadamard and the spatial derivative is in the sense of fractional Laplacian. Applying the newly customized integral transforms, i.e. the amended Laplace transform and the amended Mellin transform, the fundamental solution of the equation with $\alpha \in (0,1)$ can be obtained and its asymptotic estimates are shown. Then we study the decay estimate of the solution to the considered equation in $L^p({\mathbb{R}}^d)$ and $L^{p,\mathrm{\infty }}({\mathbb{R}}^d)$. Furthermore, gradient estimates and large time behaviour of the solution are displayed. Finally, optimal $L^2$ decay estimate of the solution are obtained by Fourier analysis techniques.

Keywords:

• Caputo–Hadamard derivative
• fractional Laplacian
• Fox H-function
• fundamental solution
• decay estimate
• asymptotic behaviour

2010 Mathematics Subject Classifications:

• 26A33
• 35R11

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was partially supported by the National Natural Science Foundation of China under Grant nos. 11872234,11926319.