The well-posedness for semilinear time fractional wave equations on $ \mathbb R^N $

Yong Zhou; Jia Wei He; Ahmed Alsaedi; Bashir Ahmad; Yong Zhou; Jia Wei He; Ahmed Alsaedi; Bashir Ahmad

doi:10.3934/era.2022151

Electronic Research Archive

2022, Volume 30, Issue 8: 2981-3003. doi: 10.3934/era.2022151

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The well-posedness for semilinear time fractional wave equations on $ \mathbb R^N $

1.
Faculty of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
2.
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3.
College of Mathematics and Information Science, Guangxi University, Nanning 530004, China

Academic Editor: Arran Fernandez

Received: 29 November 2021 Revised: 23 March 2022 Accepted: 26 May 2022 Published: 01 June 2022

This paper is concerned with the semilinear time fractional wave equations on the whole Euclidean space, also known as the super-diffusive equations. Considering the initial data in the fractional Sobolev spaces, we prove the local/global well-posedness results of $ L^2 $-solutions for linear and semilinear problems. The methods of this paper rely upon the relevant wave operators estimates, Sobolev embedding and fixed point arguments.
- fractional derivative,
- fractional wave equations,
- well-posedness
Citation: Yong Zhou, Jia Wei He, Ahmed Alsaedi, Bashir Ahmad. The well-posedness for semilinear time fractional wave equations on $ \mathbb R^N $[J]. Electronic Research Archive, 2022, 30(8): 2981-3003. doi: 10.3934/era.2022151

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Abstract

This paper is concerned with the semilinear time fractional wave equations on the whole Euclidean space, also known as the super-diffusive equations. Considering the initial data in the fractional Sobolev spaces, we prove the local/global well-posedness results of $ L^2 $-solutions for linear and semilinear problems. The methods of this paper rely upon the relevant wave operators estimates, Sobolev embedding and fixed point arguments.

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