Conditional stability in a backward Cahn-Hilliard equation via a Carleman estimate

Impact Factor:1.1

DOI number:10.1515/jiip-2017-0082

Journal:Journal of Inverse and Ill-Posed Problems

Key Words:Cahn-Hilliard equation; backward problem; Carleman estimate; conditional stability

Abstract:We consider a Cahn-Hilliard equation in a bounded domain Omega in R-n over a time interval (0, T) and discuss the backward problem in time of determining intermediate data u(x, theta), theta epsilon (0, T), x epsilon Omega from the measurement of the final data u(x, T), x epsilon Omega. Under suitable a priori boundness assumptions on the solutions u(x, t), we prove a conditional stability estimate for the semilinear Cahn-Hilliard equation parallel to u(., theta)parallel to(L2(Omega)) <= C parallel to u(., T)parallel to(kappa 0)(H2(Omega)), and a conditional stability estimate for the linear Cahn-Hilliard equation parallel to u(., theta)parallel to(H beta(Omega)) <= C parallel to u(. , T)parallel to(kappa 1)(H2(Omega)), where theta epsilon (0, T), beta epsilon (0, 4) and kappa(0), kappa(1) epsilon (0, 1). The proof is based on a Carleman estimate with the weight function e(2se lambda t) with large parameters s, lambda epsilon R^+.

Co-author:Shumin Li

First Author:Yunxia Shang

Indexed by:Journal paper

Document Code:000635601700001

Discipline:Natural Science

Document Type:J

Volume:29

Issue:2

Page Number:159-171

ISSN No.:0928-0219

Translation or Not:no

Date of Publication:2021-04-01

Included Journals:SCI、EI

Links to published journals:https://www.degruyter.com/document/doi/10.1515/jiip-2017-0082/html