Regularization of an inverse nonlinear parabolic problem with time-dependent coefficient and locally Lipschitz source term

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:449

Article

Than Nguyen Huy¹  Mach Nguyet Minh²  Kirane, Mokhtar³  Bin-Mohsin, Bandar⁴ 

[1] Ton Duc Thang Univ, Fac Math & Stat, Appl Anal Res Grp, Ho Chi Minh City, Vietnam

[2] Univ Helsinki, Dept Math & Stat, Helsinki, Finland

[3] Univ La Rochelle, Fac Sci & Technol, LaSIE, Ave M Crepeau, F-17042 La Rochelle, France

[4] King Saud Univ, Coll Sci, Dept Math, Riyadh, Saudi Arabia

关键词: Nonlinear parabolic problem;    Backward problem;    Quasi-reversibility;    Ill-posed problem;    Contraction principle;   

DOI  :  10.1016/j.jmaa.2016.11.083

来源: Elsevier

【 摘 要 】

We consider a backward problem of finding a function u satisfying a nonlinear parabolic equation in the form u(t) + a(t)Au(t) = f (t, u(t)) subject to the final condition u(T) = phi. Here A is a positive self-adjoint unbounded operator in a Hilbert space H and f satisfies a locally Lipschitz condition. This problem is ill posed. Using quasi-reversibility method, we shall construct a regularized solution u(epsilon) from the measured data alpha(epsilon) and phi(epsilon). We show that the regularized problems are well posed and that their solutions converge to the exact solutions. Error estimates of logarithmic type are given and a simple numerical example is presented to illustrate the method as well as verify the error estimates given in the theoretical parts. (C) 2016 Published by Elsevier Inc.

【 授权许可】

Free