Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator

Dinh Nguyen Duy Hai

doi:10.3934/cpaa.2022043

Article Contents

2022, Volume 21, Issue 5: 1715-1734. Doi: 10.3934/cpaa.2022043

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Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator

Dinh Nguyen Duy Hai

Department of Economic Mathematics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

Received: June 30, 2021

Revised: December 31, 2021

Early access: February 2022

Published: May 2022

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Abstract

In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the $ n $-dimensional space. We prove that the problem is ill-posed in the sense of Hadamard, i.e., the solution, if it exists, does not depend continuously on the data. To regularize the problem, we propose two modified versions of the so-called optimal filtering method of Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162–170]. According to different a priori assumptions on the regularity of the exact solution, we obtain some sharp optimal estimates of the Hölder-Logarithmic type in the Sobolev space $ H^q(\mathbb{R}^n) $.

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Mathematics Subject Classification: Primary: 35K55, 35S16, 35R25; Secondary: 47J06; Tertiary: 60H50.

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