  We apply functional analytical and variational methods in order to study
well-posedness and qualitative properties of evolution equations on product
Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of
sesquilinear mappings. We apply our results to parabolic problems of different
nature: a coupled diffusive system arising in neurobiology, a strongly damped
wave equation, a heat equation with dynamic boundary conditions, and a general
semilinear Hodgkin--Huxley sytem.
