Maximal regularity for evolution equations governed by non-autonomous forms

Abstract

We consider a non-autonomous evolutionary problem \[ \dot{u} (t)+\mathcal A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator $\mathcal A(t)\colon V\to V^\prime$ is associated with a form $\mathfrak{a}(t,.,.)\colon V\times V \to \mathbb R$ and $u_0\in V$. Our main concern is to prove well-posedness with maximal regularity, which means the following. Given a Hilbert space $H$ such that $V$ is continuously and densely embedded into $H$ and given $f\in L^2(0,T;H)$, we are interested in solutions $u \in H^1(0,T;H)\cap L^2(0,T;V)$. We do prove well-posedness in this sense whenever the form is piecewise Lipschitz-continuous and satisfies the square root property. Moreover, we show that each solution is in $C([0,T];V)$. The results are applied to non-autonomous Robin-boundary conditions and maximal regularity is used to solve a quasilinear problem.