Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations

Yongqiang Fu; Xiaoju Zhang

doi:10.3934/dcdsb.2021091

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2022, Volume 27, Issue 3: 1301-1322. Doi: 10.3934/dcdsb.2021091

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Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations

Yongqiang Fu and
Xiaoju Zhang^,

School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

^* Corresponding author: Xiaoju Zhang
^* Corresponding author: Xiaoju Zhang

Received: October 2020

Early access: March 2021

Published: March 2022

The first author is supported by the National Natural Science Foundation of China grant No.11771107.

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Abstract

In this paper, we investigate initial boundary value problems for Kirchhoff-type diffusion equations $ \partial_{t}^{\beta}u+M(\|u\|_{H_0^{s}(\Omega)}^2)(-\Delta)^{s}u = \gamma|u|^{\rho}u+g(t,x) $ with the Caputo time fractional derivatives and fractional Laplacian operators. We establish a new compactness theorem concerning time fractional derivatives. By Galerkin method, let $ 0<\rho<\frac{4s}{N-2s} $ when $ \gamma<0 $, and $ 0<\rho<\min\{\frac{4s}{N},\frac{2s}{N-2s}\} $ when $ \gamma>0 $, then we obtain the global existence and uniqueness of weak solutions for Kirchhoff problems. Furthermore, we get the decay properties of weak solutions in $ L^2(\Omega) $ and $ L^{\rho+2}(\Omega) $. Remarkably, the decay rate differs from that in the case $ \beta = 1 $.

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Mathematics Subject Classification: Primary: 35R11; Secondary: 35A01, 35B40.

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