Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation

Nguyen Huy Tuan

doi:10.3934/dcdss.2021113

Article Contents

2021, Volume 14, Issue 12: 4551-4574. Doi: 10.3934/dcdss.2021113 open access

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Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation

Nguyen Huy Tuan^1,2,

1.
Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
2.
Department of Mathematics and Computer Sciences, University of Science, (Viet Nam National University), Ho Chi Minh City, Viet Nam

Received: July 31, 2021

Revised: August 31, 2021

Early access: October 2021

Published: December 2021

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Abstract

In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source term. To overcome the intricacies of the proofs, we applied $ L^p-L^q $ estimate for biharmonic heat semigroup, Banach fixed point theory, some estimates for Mittag-Lefler functions and Wright functions, and also Sobolev embeddings. For the second result involving the Cahn-Hilliard equation with the Caputo-Fabrizio operator, we first show the local existence result. In addition, we first provide that the connections of the mild solution between the Cahn-Hilliard equation in the case $ 0 < {\alpha} < 1 $ and $ {\alpha} = 1 $. This is the first result of investigating the Cahn-Hilliard equation with this type of derivative. The main key of the proof is based on complex evaluations involving exponential functions, and some embeddings between $ L^p $ spaces and Hilbert scales spaces.

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Mathematics Subject Classification: 35R11, 35B65, 26A33.

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