A backward parabolic problem with locally Lipschitz source

We consider a backward problem of finding the function u satisfying a nonlinear parabolic equation in the form ut + Au(t) = f (t, u(t)) subject to the final condition u(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 regularized solutions uε from inexact data φε satisfying ‖φ − φ‖ → 0 as → 0. We show that the regularized problem are well-posed and that their solutions converge to the exact solutions. Error estimate is given.

[1]  Tom Fleischer,et al.  Applied Functional Analysis , 2016 .

[2]  Dang Duc Trong,et al.  On a backward parabolic problem with local Lipschitz source , 2014 .

[3]  Dang Duc Trong,et al.  A nonlinear parabolic equation backward in time: Regularization with new error estimates , 2010 .

[4]  Beth M. Campbell Hetrick,et al.  Continuous dependence on modeling for nonlinear ill-posed problems , 2009 .

[5]  Stig Larsson,et al.  Introduction to stochastic partial differential equations , 2008 .

[6]  Pham Hoang Quan,et al.  A Nonlinear Case of the 1-D Backward Heat Problem: Regularization and Error Estimate , 2007 .

[7]  Q. Zheng,et al.  Regularization for a class of ill-posed Cauchy problems , 2005 .

[8]  M. Denche,et al.  A modified quasi-boundary value method for ill-posed problems , 2005 .

[9]  Seth F. Oppenheimer,et al.  Quasireversibility Methods for Non-Well-Posed Problems , 1994 .

[10]  Nguyen Thanh Long,et al.  Approximation of a parabolic non-linear evolution equation backwards in time , 1994 .

[11]  B. Blackwell,et al.  Inverse Heat Conduction: Ill-Posed Problems , 1985 .

[12]  Ralph E. Showalter,et al.  The final value problem for evolution equations , 1974 .

[13]  L. Payne,et al.  Some general remarks on improperly posed problems for partial differential equations , 1973 .

[14]  K. Miller,et al.  Stabilized quasi-reversibilite and other nearly-best-possible methods for non-well-posed problems , 1973 .

[15]  H. Gajewski,et al.  Zur regularisierung einer Klasse nichtkorrekter probleme bei evolutionsgleichungen , 1972 .

[16]  Robert Lattès,et al.  Méthode de quasi-réversibilbilité et applications , 1967 .