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December 2021 Analysis of a quasi-reversibility method for nonlinear parabolic equations with uncertainty data
Nguyen Huy Tuan, Erkan Nane, Dang Duc Trong
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Illinois J. Math. 65(4): 793-845 (December 2021). DOI: 10.1215/00192082-9501497

Abstract

In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed—i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time dependent coefficients. The equations with constant coefficients include the heat equation, extended Fisher–Kolmogorov equation, Swift–Hohenberg equation, and many others. The equations with time dependent coefficients include Fisher-type logistic equations, the Huxley equation, and the Fitzhugh–Nagumo equation. The methods developed in this paper can also be applied to get approximate solutions to several other equations including the 1-D Kuramoto–Sivashinsky equation, 1-D modified Swift–Hohenberg equation, strongly damped wave equation, and 1-D Burgers equation with randomly perturbed operator. Some numerical examples are given which illustrate the effectiveness of our method.

Citation

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Nguyen Huy Tuan. Erkan Nane. Dang Duc Trong. "Analysis of a quasi-reversibility method for nonlinear parabolic equations with uncertainty data." Illinois J. Math. 65 (4) 793 - 845, December 2021. https://doi.org/10.1215/00192082-9501497

Information

Received: 28 December 2020; Revised: 28 July 2021; Published: December 2021
First available in Project Euclid: 2 December 2021

Digital Object Identifier: 10.1215/00192082-9501497

Subjects:
Primary: 35K05
Secondary: 35K99 , 47H10 , 47J06

Rights: Copyright © 2021 by the University of Illinois at Urbana–Champaign

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Vol.65 • No. 4 • December 2021
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