Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case

This paper is concerned with the asymptotic behaviors of solution to time–space fractional partial differential equation with Caputo–Hadamard derivative (in time) and fractional Laplacian (in space) in the hyperbolic case, that is, the Caputo–Hadamard derivative order \begin{document}$ \alpha $\end{document} lies in \begin{document}$ 1<\alpha<2 $\end{document} . In view of the technique of integral transforms, the fundamental solutions and the exact solution of the considered equation are derived. Furthermore, the fundamental solutions are estimated and asymptotic behaviors of its analytical solution is established in \begin{document}$ L^{p}(\mathbb{R}^{d}) $\end{document} and \begin{document}$ L^{p,\infty} (\mathbb{R}^{d}) $\end{document} . We finally investigate gradient estimates and large time behavior for the solution.