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=1κ ajDj for each 0 ≤ t ≤ T with aj ∈ 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 Lp(ℝ), 1 ≤ p < ∞, with time-dependent coefficients.
All Science Journal Classification (ASJC) codes
- Applied Mathematics