Maximal regularity for second order non-autonomous Cauchy problems
Tom 189 / 2008
Studia Mathematica 189 (2008), 205-223 MSC: Primary 47E05; Secondary 34G10, 35B65, 47D09. DOI: 10.4064/sm189-3-1
We consider some non-autonomous second order Cauchy problems of the form $$ \ddot u + B(t) \dot u + A(t) u = f \quad (t\in [0,T]) , \ \quad u(0) = \dot u (0) =0. $$ We assume that the first order problem $$ \dot u + B(t) u = f \quad (t\in [0,T]) , \ \quad u(0) =0, $$ has $L^p$-maximal regularity. Then we establish $L^p$-maximal regularity of the second order problem in situations when the domains of $B(t_1)$ and $A(t_2)$ always coincide, or when $A(t) = \kappa B(t)$.