Maximal regularity for the damped wave equations

Abstract

We consider the problem of maximal regularity for non-autonomous second order Cauchy problems

$$\begin{aligned} \left\{ \begin{array}{l} u''(t)+{\mathcal {B}}(t)u'(t)+{\mathcal {A}}(t)u(t)=f(t) \quad t\text{-a.e.} \\ u(0)=u_{0},\quad u'(0)=u_{1}. \end{array} \right. \end{aligned}$$

Here, the time dependent operator ${\mathcal {A}}(t)$ is bounded from the Hilbert space ${\mathcal {V}}$ to its dual space ${\mathcal {V}}'$ and ${\mathcal {B}}(t)$ is associated with a sesquilinear form $\mathfrak {b}(t,\cdot ,\cdot )$ with domain ${\mathcal {V}}$. We prove maximal $L^p$-regularity results and other regularity properties for the solutions of the above equation under minimal regularity assumptions on the operators. Our result is motivated by boundary value problems.