MR3052745 Atsushi YAGI Atsushi YAGI Maximal Regularity of Hölder Type for Abstract Parabolic Evolution Equations Funkcialaj Ekvacioj. Serio Internacia 55 2012 405--428 http://fe.math.kobe-u.ac.jp/FE/FullPapers/55-3/55_405.pdf http://www.ams.org/mathscinet-getitem?mr=MR3052745 This paper shows Hölder type maximal regularity for the Cauchy problem of linear abstract parabolic evolution equations in Banach spaces. In the monograph [20, Chapter 3], such regularity has already been established in the case when the domains of the linear operators appearing in the equations are temperately varying with respect to the time variable. This paper is devoted to handling the more general case when the domains are completely varying. Maximal regularity is known to be an essential property of linear parabolic differential equations. Abstract parabolic equation, Maximal regularity. 35K90, 35B30. 55-405 2012 Maximal Regularity of Hölder Type for Abstract Parabolic Evolution Equations Atsushi YAGI Atsushi YAGI
1 Da Prato, G.; Grisvard, P. Sommes d'opérateurs linéaires et équations différentielles opérationnelles J. Math. Pures Appl. (9) 54 1975 305-387 MR0442749
2 Da Prato, G.; Grisvard, P. Maximal regularity for evolution equations by interpolation and extrapolation J. Funct. Anal. 58 1984 107-124 MR0757990
3 Favaron, A. Optimal time and space regularity for solutions of degenerate differential equations Cent. Eur. J. Math. 7 2009 249-271 MR2506965
4 Favini, A.; Lorenzi, A.; Tanabe, H.; Yagi, A. An $L_p$-approach to singular linear parabolic equations in bounded domains Osaka J. Math. 42 2005 385-406 MR2147730
5 Favini, A.; Yagi, A. Space and time regularity for degenerate evolution equations J. Math. Soc. Japan 44 1992 331-350 MR1154846
6 Favini, A.; Yagi, A. Multivalued linear operators and degenerate evolution equations Ann. Mat. Pura Appl. (4) 163 1993 353-384 MR1219605
7 Favini, A.; Yagi, A. Degenerate Differential Equations in Banach Spaces Monographs and Textbooks in Pure and Applied Mathematics, 215, Marcel Dekker, Inc., New York 1999 MR1654663
8 Guidotti, P. Optimal regularity for a class of singular abstract parabolic equations J. Differential Equations 232 2007 468-486 MR2286388
9 Krein, S. G. Linear Differential Equations in a Banach Space Izdat. "Nauka", Moscow, 1967 (in Russian). English translation, Translations of Mathematical Monographs, Vol. 29, American Mathematical Society, Providence, R.I. 1971 MR0247239 10 Lunardi, A. Analytic Semigroups and Optimal Regularity in Parabolic Problems Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel 1995 MR1329547
11 Osaki, K.; Yagi, A. Global existence for a chemotaxis-growth system in $\mathbb R^2$ Adv. Math. Sci. Appl. 12 2002 587-606 MR1943982
12 Tanabe, H. Note on singular perturbation for abstract differential equations Osaka J. Math. 1 1964 239-252 MR0172142
13 Tanabe, H. Equations of Evolution Iwanami, Tokyo, 1975 (in Japanese). English translation, Monographs and Studies in Mathematics, 6, Pitman, Boston, Mass.-London 1979 MR0533824 14 Tanabe, H. Functional Analytic Methods for Partial Differential Equations Monographs and Textbooks in Pure and Applied Mathematics, 204, Marcel Dekker, Inc., New York 1997 MR1413304
15 Watanabe, M. A remark on fractional powers of linear operators in Banach spaces Proc. Japan Acad. Ser. A Math. Sci. 53 1977 4-7 MR0454694
16 Weber, F. On products of non-commuting sectorial operators Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 1998 499-531 MR1677994
17 Yagi, A. On the abstract linear evolution equations in Banach spaces J. Math. Soc. Japan 28 1976 290-303 MR0397478
18 Yagi, A. Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups Funkcial. Ekvac. 32 1989 107-124 MR1006090 1006090 19 Yagi, A. Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, II Funkcial. Ekvac. 33 1990 139-150 MR1065472 1065472 20 Yagi, A. Abstract Parabolic Evolution Equations and their Applications Springer Monographs in Mathematics. Springer-Verlag, Berlin 2010 MR2573296