## Abstract

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)=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 u_{ε} from the measured data a_{ε} and φ_{ε}. 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.

Original language | British English |
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Pages (from-to) | 697-717 |

Number of pages | 21 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 449 |

Issue number | 1 |

DOIs | |

State | Published - 1 May 2017 |

## Keywords

- Backward problem
- Contraction principle
- Ill-posed problem
- Nonlinear parabolic problem
- Quasi-reversibility