Maximal Regularity for Non-Autonomous Second Order Cauchy Problems

We consider non-autonomous wave equations $$\{\begin{array}{r}{\displaystyle \mathrm{\&}\ddot{u}(t)+\text{B}(t)\dot{u}(t)+\text{A}(t)u(t)=f(t)t\text{-a.e.}}\\ {\displaystyle \mathrm{\&}u(0)=u_0,\dot{u}(0)=u_1\mathrm{.}}\end{array}$$ where the operators $\text{A}(t)$ and $\text{B}(t)$ are associated with time-dependent sesquilinear forms $\text{fra}(t,\mathrm{.},\mathrm{.})$ and $\text{frb}$ 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 $V^{\mathrm{\prime }}$ and $H$. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem.