Maximal Regularity for Non-Autonomous Second Order Cauchy Problems

Abstract

We consider non-autonomous wave equations \[ \left\{ \begin{aligned} \&\ddot u(t) + \B(t)\dot u(t) + \A(t)u(t) = f(t) \quad t\text{-a.e.}\\ \&u(0)=u_0,\, \dot u(0) = u_1. \end{aligned} \right. \] where the operators $\A(t)$ and $\B(t)$ are associated with time-dependent sesquilinear forms $\fra(t,.,.)$ and $\frb$ defined on a Hilbert space $H$ with the same domain $V$. The initial values satisfy $u_0\in V$ and $u_1\in H$. We prove well-posedness and maximal regularity for the solution both in the spaces $V^{\prime }$ and $H$. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem.