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

Changpin Li; Zhiqiang Li

doi:10.3934/dcdss.2021023

Article Contents

2021, Volume 14, Issue 10: 3659-3683. Doi: 10.3934/dcdss.2021023

This issue Previous Article Feedback stabilization of bilinear coupled hyperbolic systems Next Article Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves

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

Changpin Li^, and
Zhiqiang Li

Department of Mathematics, Shanghai University, 200444, Shanghai, China

^* Corresponding author: Changpin Li
^* Corresponding author: Changpin Li

Received: April 2020

Revised: August 2020

Early access: March 2021

Published: October 2021

The first author is supported by NSFC grant 11872234 and 11926319.

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Abstract

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 $ \alpha $ lies in $ 1<\alpha<2 $. 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 $ L^{p}(\mathbb{R}^{d}) $ and $ L^{p,\infty} (\mathbb{R}^{d}) $. We finally investigate gradient estimates and large time behavior for the solution.

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Mathematics Subject Classification: Primary: 26A33; Secondary: 35R11.

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