Global existence and long-time behavior of solutions for fully nonlocal Boussinesq equations

Xiaoju Zhang; Kai Zheng; Yao Lu; Huanhuan Ma; Xiaoju Zhang; Kai Zheng; Yao Lu; Huanhuan Ma

doi:10.3934/era.2023274

Electronic Research Archive

2023, Volume 31, Issue 9: 5406-5424. doi: 10.3934/era.2023274

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Global existence and long-time behavior of solutions for fully nonlocal Boussinesq equations

1.
College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China
2.
Henan Key Laboratory of Educational Artificial Intelligence and Personalized Learning, Henan Province, Xinxiang 453007, China
3.
Network Center, Henan Normal University, Xinxiang 453007, China
4.
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China
5.
School of Environment, Henan Normal University, Xinxiang 453007, China

Received: 26 May 2023 Revised: 12 July 2023 Accepted: 24 July 2023 Published: 31 July 2023

In this paper, we study initial boundary value problems for the following fully nonlocal Boussinesq equation

$ _0^{C}D_{t}^{\beta}u+(-\Delta)^{\sigma}u+(-\Delta)^{\sigma}{_0^{C}D_{t}^{\beta}}u = -(-\Delta)^{\sigma}f(u) $

with spectral fractional Laplacian operators and Caputo fractional derivatives. To our knowledge, there are few results on fully nonlocal Boussinesq equations. The main difficulty is that each term of this equation has nonlocal effect. First, we obtain explicit expressions and some rigorous estimates of the Green operators for the corresponding linear equation. Further, we get global existence and some decay estimates of weak solutions. Second, we establish new chain and Leibnitz rules concerning $ (-\Delta)^{\sigma} $. Based on these results and small initial conditions, we obtain global existence and long-time behavior of weak solutions under different dimensions $ N $ by Banach fixed point theorem.
- Boussinesq equation,
- fractional operator,
- global existence,
- long-time behavior
Citation: Xiaoju Zhang, Kai Zheng, Yao Lu, Huanhuan Ma. Global existence and long-time behavior of solutions for fully nonlocal Boussinesq equations[J]. Electronic Research Archive, 2023, 31(9): 5406-5424. doi: 10.3934/era.2023274

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Abstract

In this paper, we study initial boundary value problems for the following fully nonlocal Boussinesq equation

$ _0^{C}D_{t}^{\beta}u+(-\Delta)^{\sigma}u+(-\Delta)^{\sigma}{_0^{C}D_{t}^{\beta}}u = -(-\Delta)^{\sigma}f(u) $

with spectral fractional Laplacian operators and Caputo fractional derivatives. To our knowledge, there are few results on fully nonlocal Boussinesq equations. The main difficulty is that each term of this equation has nonlocal effect. First, we obtain explicit expressions and some rigorous estimates of the Green operators for the corresponding linear equation. Further, we get global existence and some decay estimates of weak solutions. Second, we establish new chain and Leibnitz rules concerning $ (-\Delta)^{\sigma} $. Based on these results and small initial conditions, we obtain global existence and long-time behavior of weak solutions under different dimensions $ N $ by Banach fixed point theorem.

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