Semilinear parabolic equations involving measures and low regularity data

Abstract

A detailed study of abstract semilinear evolution equations of the form u ˙ + A u = μ ( u ) \dot u+Au=\mu (u) is undertaken, where − A -A generates an analytic semigroup and μ ( u ) \mu (u) is a Banach space valued measure depending on the solution. Then it is shown that the general theorems apply to a variety of semilinear parabolic boundary value problems involving measures in the interior and on the boundary of the domain. These results extend far beyond the known results in this field. A particularly new feature is the fact that the measures may depend nonlinearly and possibly nonlocally on the solution.