### Abstract

We prove regularization for a certain ill-posed parabolic evolution problem in a Banach space X by obtaining Hölder-continuous dependence of its solution on modeling. In particular, we consider the generally ill-posed problem du/dt = A(t, D)u(t), 0 ≤ s ≤ t < T, with initial data u(s) = χ, in X where -D is the infinitesimal generator of a bounded holomorphic semigroup of angle θ ∈ (π/4, π/2]on X, and A(t, D) = ∑ _{j=1}^{κ} a_{j}D^{j} for each 0 ≤ t ≤ T with a_{j} ∈ C([0, T] : ℝ^{+}) for 1 ≤ j ≤ κ. Assuming there exists a solution u(t) of the problem adhering to certain stabilizing conditions, we approximate u(t) by the solution of an approximate well-posed problem. We then use this estimate to prove the existence of a family of regularizing operators for the given ill-posed problem. The theory has applications to the backwards heat equation and other ill-posed partial differential equations in L^{p}(ℝ), 1 ≤ p < ∞, with time-dependent coefficients.

Original language | English (US) |
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Pages (from-to) | 667-699 |

Number of pages | 33 |

Journal | Journal of Inverse and Ill-Posed Problems |

Volume | 20 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - Dec 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics