... 1964.15 A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16 of Progress inNonlinear Differential Equations and their Applications,Birkh¨auser, Basel, Switzerland, ... Ibrahim, M. Majdoub, and N. Masmoudi, “Double logarithmic inequality with a sharp constant,”Proceedings of the American Mathematical Society, vol. 135, no. 1, pp. 87–97, 2007.Boundary Value ... 606-8502, Japan4Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku,Osaka 558-8585, JapanCorrespondence should be addressed to Yoshihiro Sawano, yoshihiro-sawano@celery.ocn.ne.jpReceived...