Nonautonomous Parabolic Equations Involving Measures

where Ω ⊂ R and Γ is the boundary of Ω. In this case μΩ is a bounded Radon measure on Ω× J and μΓ is such a measure on Γ× J . In [10] the semilinear analogues of (0.1) and (0.2), where μ and (μΩ, μΓ), respectively, may also depend nonlinearly and nonlocally on the unknown solution u, are investigated. Based on the results of [5], general existence, uniqueness, and continuity theorems are proved and it is shown that they have numerous applications to semilinear parabolic problems (also see [9] for applications to control problems). Formally, the solution of (0.1) is given by U ? μ, where U is the semigroup generated by −A. The difficulty lies in defining a convolution of U with a Banach space valued measure possessing suitable regularity properties which reflect the parabolicity of problem (0.1). In [5] this problem is resolved by a detailed study of semigroups in spaces of integrable functions, and a duality approach. In [10, Section 16] it is shown how the results of [5] can be extended to the nonautonomous case. Nonautonomous equations arise naturally in linearization procedures. In this connection it is important that only minimal regularity conditions are required. (See [10, Section 19], where the