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We study stability and integrability of linear non-autonomous evolutionary Cauchy-problem $$\left(\mathrm{P}\right)\left\{\begin{array}{c}\dot{u}\left(t\right)+A\left(t\right)u\left(t\right)=f\left(t\right)t\text{-a.e.}\text{on}[0,\tau ],u\left(0\right)=0,\hfill \end{array}\right.$$ where $A:[0,\tau ]\to ℒ(X,D)$ is a bounded and strongly measurable function and $X$, $D$ are Banach spaces such that $D\underset{d}{\to }\hookrightarrow X$. Our main concern is to characterize $L^p$-maximal regularity and to give an explicit approximation of the problem (P).

We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded $H^{\infty }$-calculus and is based on elementary analysis.