Nonlocal biharmonic evolution equations with Dirichlet and Navier boundary conditions

Kehan Shi; Ying Wen

doi:10.3934/dcdsb.2022089

Article Contents

2023, Volume 28, Issue 1: 560-579. Doi: 10.3934/dcdsb.2022089

This issue Previous Article On the fractional chemotaxis Navier-Stokes system in the critical spaces Next Article On the asymptotic stability of a Bresse system with two fractional damping terms: Theoretical and numerical analysis

Nonlocal biharmonic evolution equations with Dirichlet and Navier boundary conditions

Kehan Shi^1, , and
Ying Wen^2,

1.
Department of Mathematics, China Jiliang University, Hangzhou 310018, China
2.
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

^*Corresponding author: Kehan Shi
^*Corresponding author: Kehan Shi

Received: December 2021

Revised: April 2022

Early access: May 11, 2022

Published online: May 11, 2022

This work was supported by the Natural Science Foundation of Zhejiang Province, China (LQ21A010010) and the National Natural Science Foundation of China (12001509).

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Abstract

This paper studies a nonlocal biharmonic evolution equation with Dirichlet boundary condition that arises in image restoration. We prove the existence and uniqueness of solutions to the nonlocal problem by the variational method and show that the solutions of the nonlocal problem converge to the solution of the classical biharmonic equation with Dirichlet boundary condition if the nonlocal kernel is rescaled appropriately. The asymptotic behavior is discussed. Besides, we study the Navier problem by transforming it into a Dirichlet problem with a fixed point. The existence, uniqueness, convergence under the rescaling of the kernel, and asymptotic behavior of solutions to the Navier problem are discussed.

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Mathematics Subject Classification: 45A05, 45M05, 35G16.

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