## Abstract

We prove regularization for ill-posed evolution problems that are both inhomogeneous and nonautonomous in a Hilbert Space H. We consider the ill-posed problem du=dt = A(t;D)u(t) + h(t), u(s) = χ, 0 ≤ s ≤ t < T where A(t;D) = Σ _{j=1}^{k} a_{j} (t)D^{j} with a_{j} ∈ C([0; T] : ℝ^{+}) for each 1 ≤ j ≤ k and D a positive, self-adjoint operator in H. Assuming there exists a solution u of the problem with certain stabilizing conditions, we approximate u by the solution ν_{β} of the approximate well-posed problem dν=dt = f_{β} (t;D)ν(t)+h(t), ν(s) = χ, 0 ≤ s ≤ t < T where 0 < β < 1. Our method implies the existence of a family of regularizing operators for the given ill-posed problem with applications to a wide class of ill-posed partial differential equations including the inhomogeneous backward heat equation in L^{2}(ℝ^{n}) with a time-dependent diffusion coefficient.

Original language | English (US) |
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Pages (from-to) | 259-272 |

Number of pages | 14 |

Journal | Discrete and Continuous Dynamical Systems - Series S |

Issue number | SUPPL. |

State | Published - Nov 2013 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics