Maximal regularity for second order non-autonomous Cauchy problems

We consider some non-autonomous second order Cauchy problems of the form ü + B(t)̇ + A(t)u = f (t ∈ [0, T]), u(0) = ̇(0) = 0. We assume that the first order problem ü + B(t)u = f (t∈ [0, T]), u(0) = 0, has L^p-maximal regularity. Then we establish L^p-maximal regularity of the second order problem in situations when the domains of B(t<inf>1</inf>) and A(t<inf>2</inf>) always coincide, or when A(t) = <inf>κ</inf>B(t). © Instytut Matematyczny PAN, 2008.