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

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• 作者：Nguyen Huy Tuan^a ; ^{nguyenhuytuan@tdt.edu.vn} ; Nguyet Minh Mach^b ; ^{minh.mach@helsinki.fi} ; Mokhtar Kirane^c ; ^{mokhtar.kirane@univ-lr.fr} ; Bandar Bin-Mohsin^d ; ^{balmohsen@ksu.edu.sa}
• 关键词：Nonlinear parabolic problem ; Backward problem ; Quasi-reversibility ; Ill-posed problem ; Contraction principle
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 May 2017
• 年：2017
• 卷：449
• 期：1
• 页码：697-717
• 全文大小：1236 K
• 卷排序：449

文摘

We consider a backward problem of finding a function u   satisfying a nonlinear parabolic equation in the form ut+a(t)Au(t)=f(t,u(t))ut+a(t)Au(t)=f(t,u(t)) subject to the final condition u(T)=φu(T)=φ. 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 uεuε from the measured data aεaε and φεφε. 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.