Stability for evolution equations governed by a non-autonomous form

Abstract

This paper deals with the approximation of non-autonomous evolution equations of the form \begin{equation*}\label{Abstract equation} \dot u(t)+A(t)u(t)=f(t)\ \ t\in[0,T],\ \ u(0)=u_0. \end{equation*} where $A(t),\ t\in [0,T]$ arise from a non-autonomous sesquilinear forms $\mathfrak a(t;\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ Assuming the existence of a sequence $\mathfrak a_n:[0,T]\times V\times V\longrightarrow\mathbb C, n\in \mathbb N$ of non-autonomous forms such that the associated Cauchy problem has $L^2$-maximal regularity in $H$ and $\mathfrak a_n(t,u,v)$ converges to $\mathfrak a(t,u,v)$ as $n\to \infty,$ then among others we show under additional assumptions that the limit problem has $L^2$-maximal regularity. Further we show that the convergence is uniformly on the initial data $u_0$ and the inhomogeneity $f.$