On a non-local Kirchhoff type equation with random terminal observation

Phuong Nguyen Duc; Tien Nguyen Van; Tuan Nguyen Anh

doi:10.3934/dcdss.2023109

Article Contents

2024, Volume 17, Issue 3: 1011-1027. Doi: 10.3934/dcdss.2023109

This issue Previous Article Existence results for fractional evolution equations with superlinear growth nonlinear terms Next Article New regularization and error estimate on terminal value problem for elliptic equations

On a non-local Kirchhoff type equation with random terminal observation

1.
Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Vietnam
2.
Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam
3.
Vietnam National University, Ho Chi Minh City, Vietnam
4.
Department of Mathematics, FPT University, Hanoi, Vietnam
5.
Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam
6.
Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam

^*Corresponding author: Tuan Nguyen Anh
^*Corresponding author: Tuan Nguyen Anh

Received: March 2023

Early access: May 29, 2023

Published online: May 29, 2023

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Abstract

In this work, we are concerned with the terminal value problem for the time fractional equation (in the sense of Conformable fractional derivative) with a nonlocal term of the Kirchhoff type

$ \partial_t^\alpha u = K\Big(\|\nabla u\|_{L^2(\mathcal{D})}\Big)\Delta u + f(x,t), \quad (x,t) \in (0,T)\times \mathcal{D} $

subject to the final data which is blurred by random Gaussian white noise. The principal goal of this article is to recover the solution $ u $. This problem is severely ill-posed in the sense of Hadamard, because of the violation of the continuous dependence of the solution on the data (the solution's behavior does not change continuously with the final condition). By applying non-parametric estimates of the value data from observation data and the truncation method for the Fourier series, we obtain a regularized solution. Under some priori assumptions, we derive an error estimate between a mild solution and its regularized solution.

Keywords:

Mathematics Subject Classification: Primary: 26A33, 35K15; Secondary: 35B40, 33E12, 44A20.

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