Non-autonomous Miyadera perturbations

Abstract

We consider time dependent perturbations $B(t)$ of a non--autonomous Cauchy problem $\dot{v}(t)=A(t)v(t)$ $(CP)$ on a Banach space $X$. The existence of a mild solution $u$ of the perturbed problem is proved under Miyadera type conditions on $B(\cdot)$. In the parabolic case and $X=L^d(\Omega)$, $1 < d < \infty$, we show that $u$ is differentiable a.e. and satisfies $\dot{u}(t)=(A(t)+B(t))u(t)$ for a.e. $t$. Our approach uses perturbation results due to one of the authors, [29], and S. Monniaux and J. Prüß, [14], which are applied to the evolution semigroup induced by the evolution family related to $(CP)$. As an application we obtain solutions of a second order parabolic equation with singular lower order coefficients.