On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type

Nguyen Huy Tuan

doi:10.3934/dcdsb.2020354

Article Contents

2021, Volume 26, Issue 10: 5465-5494. Doi: 10.3934/dcdsb.2020354

This issue Previous Article Norm inflation for the Boussinesq system Next Article High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation

On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type

Nguyen Huy Tuan^,

Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam, Department of Mathematics and Computer Science, University of Science Ho Chi Minh City-VNU, Vietnam

^* Corresponding author: Nguyen Huy Tuan
^* Corresponding author: Nguyen Huy Tuan

Dedicated to Tomás Caraballo on his 60th birthday.

Received: June 2020

Revised: October 2020

Early access: December 2020

Published online: December 2020

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Abstract

We study for nonlinear Kirchhoff's model of pseudo parabolic type by considering its two different problems.

$ \bullet $ For initial value problem, we obtain the results on the existence and regularity of solutions. Moreover, we also prove that the solutions $ u $ corresponding with $ \beta < 1 $ of the problem convergence to $ u $ for $ \beta = 1 $.

$ \bullet $ For final value problem, we show that the ill-posed property in the sense of Hadamard is occurring. Using the Fourier truncation method to regularize the problem. We establish some stability estimates in the $ H^1 $ and $ L^p $ norms under some a-priori conditions on the sought solution.

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Mathematics Subject Classification: 35K55, 35K70, 35K92, 47A52, 47J06.

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