
Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case
Author(s) -
Changpin Li,
Zhiqiang Li
Publication year - 2021
Publication title -
discrete and continuous dynamical systems. series s
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 34
eISSN - 1937-1632
pISSN - 1937-1179
DOI - 10.3934/dcdss.2021023
Subject(s) - mathematics , hadamard transform , order (exchange) , space (punctuation) , laplace operator , fractional calculus , hyperbolic function , pure mathematics , mathematical analysis , combinatorics , linguistics , philosophy , finance , economics
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.