Determination of initial data for a reaction-diffusion system with variable coefficients

Vo Van Au; Mokhtar Kirane; Nguyen Huy Tuan

doi:10.3934/dcds.2019032

Article Contents

2019, Volume 39, Issue 2: 771-801. Doi: 10.3934/dcds.2019032

This issue Previous Article Free energy in a mean field of Brownian particles Next Article Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms

Determination of initial data for a reaction-diffusion system with variable coefficients

1.
Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh 700000, Vietnam
2.
LaSIE, Facult des Sciences etTechnologies, Universi de La Rochelle, Avenue M. Crpeau, La Rochelle, Cedex17042, France
3.
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

* Corresponding author
* Corresponding author

Received: October 31, 2017

Revised: June 30, 2018

Published: January 2019

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Abstract

In this paper, we study a final value problem for a reaction-diffusion system with time and space dependent diffusion coefficients. In general, the inverse problem of identifying the initial data is not well-posed, and herein the Hadamard-instability occurs. Applying a new version of a modified quasi-reversibility method, we propose a stable approximate (regularized) problem. The existence, uniqueness and stability of the corresponding regularized problem are obtained. Furthermore, we also investigate the error estimate and show that the approximate solution converges to the exact solution in $L_2$ and $\stackrel{0}{H_1}$ norms. Our method can be applied to some concrete models that arise in biology, chemical engineering, etc.

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Mathematics Subject Classification: 35K05, 35K57, 35K99, 47J06, 47H10.

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