The well-posedness of a parabolic non-autonomous semilinear equation

Abstract

In this paper, we consider a class of semilinear non-autonomous equations, where the nonlinear term does not perfectly satisfy the Lipschitz condition. It means that the nonlinear function is defined as the composition of a global Lipschitz function and non-autonomous unbounded linear operators. This case is beyond the scope of standard Cauchy-Lipshitz theory and requires further treatment. First, we deal with the well-posedness of non-autonomous parabolic equations, from the point of view of maximal regularity. This allows, indirectly, to use a fixed point theorem to prove the result. Second, we will rely on the concept of Lebesgue extensions of non-autonomous observation operators to prove the admissibility of observation for semilinear non-autonomous systems.