  This paper is devoted to the study of $L^p$-maximal regularity for
non-autonomous linear evolution equations of the form
\begin{equation*}\label{Multi-pert1-diss-non}
  \dot u(t)+A(t)B(t)u(t)=f(t)\ \ t\in[0,T],\ \
  u(0)=u_0.
  \end{equation*} where $\{A(t),\ t\in [0,T]\}$ is a family of linear unbounded
operators whereas the operators $\{B(t),\ t\in [0,T]\}$ are bounded and
invertible. In the Hilbert space situation we consider operators $A(t), \
t\in[0,T],$ which arise from sesquilinear forms. The obtained results are
applied to parabolic linear differential equations in one spatial dimension.
