Tikhonov regularization of a nonhomogeneous first-order evolution equation

Behzad Djafari Rouhani; Mohsen Rahimi Piranfar

doi:10.3934/dcdss.2023107

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2024, Volume 17, Issue 5&6: 2173-2185. Doi: 10.3934/dcdss.2023107

This issue Previous Article Bounds for the gradient of the transition kernel for elliptic operators with unbounded diffusion, drift and potential terms Next Article Shooting from singularity to singularity and a quasilinear p-Laplace-Beltrami equation with indefinite weight

Tikhonov regularization of a nonhomogeneous first-order evolution equation

Behzad Djafari Rouhani^1, , and
Mohsen Rahimi Piranfar^2,

1.
Department of Mathematical Sciences, University of Texas at El Paso, 500 W. University Ave., El Paso, TX 79968, USA
2.
Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45195-1159, Zanjan, Iran

^*Corresponding author: Behzad Djafari Rouhani
^*Corresponding author: Behzad Djafari Rouhani

Dedicated to Professor Jerome A. Goldstein’s 80th Birthday

Received: September 2022

Revised: March 2023

Early access: May 2023

Published: May 2024

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Abstract

We consider a nonhomogeneous first-order evolution equation governed by a maximal monotone operator $ A $ in a Hilbert space in the presence of a Tikhonov regularization term. We study the existence and strong convergence of weak solutions to such systems. With boundedness conditions on the central path or on the solution trajectory, without assuming the zero set of $ A $ to be nonempty and with suitable assumptions on the Tikhonov regularization coefficient, we prove that the weak solutions to the system converge strongly to the element of least norm in the zero set of $ A $. As a consequence, we provide sufficient conditions where the boundedness of the weak solutions to the nonhomogeneous Tikhonov system is equivalent to the zero set of $ A $ to be nonempty. Our work is motivated by [4,17,22] and extends some results by those authors.

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Mathematics Subject Classification: Primary: 34G25, 47J35, 47H05; Secondary: 37N40, 46N10.

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