Regularization for ill-posed inhomogeneous evolution problems in a Hilbert Space

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 L²(ℝⁿ) with a time-dependent diffusion coefficient.