Maximal regularity for second order non-autonomous Cauchy problems

Charles J. K. Batty ¹   ; Ralph Chill ²   ; Sachi Srivastava ³

¹ St. John's College Oxford OX1 3JP, Great Britain
² Laboratoire de Mathématiques et Applications de Metz – CNRS Université Paul Verlaine – Metz UMR 7122, Bât. A, Île du Saulcy 57045 Metz Cedex 1, France
³ Department of Mathematics Indian Institute of Science Bangalore 560 012, India Current address: Department of Mathematics University of Delhi Delhi, India

Studia Mathematica, Tome 189 (2008) no. 3, pp. 205-223

Résumé

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 $A(t) = \kappa B(t)$.

Détail

DOI : 10.4064/sm189-3-1

Keywords: consider non autonomous second order cauchy problems form ddot dot quad quad dot assume first order problem dot quad quad has p maximal regularity establish p maximal regularity second order problem situations domains always coincide kappa