Maximal Regularity for Fractional Cauchy Equation in Hölder Space and Its Approximation

References

[1] A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys. 280 (2015), 424–438. 10.1016/j.jcp.2014.09.031Search in Google Scholar

[2] A. Ashyralyev, A note on fractional derivatives and fractional powers of operators, J. Math. Anal. Appl. 357 (2009), no. 1, 232–236. 10.1016/j.jmaa.2009.04.012Search in Google Scholar

[3] A. Ashyralyev, C. Cuevas and S. Piskarev, On well-posedness of difference schemes for abstract elliptic problems in Lp⁢([0,T];E) spaces, Numer. Funct. Anal. Optim. 29 (2008), no. 1–2, 43–65. 10.1080/01630560701872698Search in Google Scholar

[4] A. Ashyralyev, M. Martinez, J. Pastor and S. Piskarev, On Well-Posedness of Abstract Hyperbolic Problems in Function Spaces, Further Progress in Analysis, World Scientific Publishing, Hackensack (2009), 679–688. 10.1142/9789812837332_0063Search in Google Scholar

[5] A. Ashyralyev, J. Pastor, S. Piskarev and H. A. Yurtsever, Second order equations in functional spaces: Qualitative and discrete well-posedness, Abstr. Appl. Anal. (2015), Article ID 948321. 10.1155/2015/948321Search in Google Scholar

[6] A. Ashyralyev, S. Piskarev and L. Weis, On well-posedness of difference schemes for abstract parabolic equations in Lp⁢([0,T];E) spaces, Numer. Funct. Anal. Optim. 23 (2002), no. 7–8, 669–693. 10.1081/NFA-120016264Search in Google Scholar

[7] A. Ashyralyev and P. E. Sobolevskiĭ, Well-Posedness of Parabolic Difference Equations, Oper. Theory Adv. Appl. 69, Birkhäuser, Basel, 1994. 10.1007/978-3-0348-8518-8Search in Google Scholar

[8] E. G. Bajlekova, Fractional evolution equations in Banach spaces, Dissertation, Eindhoven University of Technology, Eindhoven, 2001. Search in Google Scholar

[9] S. Bu and G. Cai, Solutions of second order degenerate integro-differential equations in vector-valued function spaces, Sci. China Math. 56 (2013), no. 5, 1059–1072. 10.1007/s11425-012-4491-ySearch in Google Scholar

[10] S. Bu and G. Cai, Well-posedness of second-order degenerate differential equations with finite delay, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 2, 349–360. 10.1017/S0013091516000262Search in Google Scholar

[11] S. Bu and G. Cai, Well-posedness of second-order degenerate differential equations with finite delay in vector-valued function spaces, Pacific J. Math. 288 (2017), no. 1, 27–46. 10.2140/pjm.2017.288.27Search in Google Scholar

[12] S. Bu and G. Cai, Well-posedness of fractional degenerate differential equations with finite delay on vector-valued functional spaces, Math. Nachr. 291 (2018), no. 5–6, 759–773. 10.1002/mana.201600502Search in Google Scholar

[13] Q. Cao, J. Pastor, S. Piskarev and S. Siegmund, Approximations of parabolic equations at the vicinity of hyperbolic equilibrium point, Numer. Funct. Anal. Optim. 35 (2014), no. 10, 1287–1307. 10.1080/01630563.2014.884580Search in Google Scholar

[14] A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim. 27 (2006), no. 7–8, 785–829. 10.1080/01630560600882723Search in Google Scholar

[15] P. Clément, G. Gripenberg and S.-O. Londen, Regularity properties of solutions of fractional evolution equations, Evolution Equations and Their Applications in Physical and Life Sciences Bad Herrenalb 1998, Lecture Notes in Pure and Appl. Math. 215, Dekker, New York (2001), 235–246. 10.1201/9780429187810-19Search in Google Scholar

[16] P. Clément and S.-O. Londen, Regularity aspects of fractional evolution equations, Rend. Istit. Mat. Univ. Trieste 31 (2000), no. 2, 19–30. Search in Google Scholar

[17] B. Eberhardt and G. Greiner, Baillon’s theorem on maximal regularity, Acta Appl. Math. 27 (1992), no. 1–2, 47–54. 10.1007/978-94-017-2721-1_5Search in Google Scholar

[18] Z. Fan, Q. Dong and G. Li, Almost exponential stability and exponential stability of resolvent operator families, Semigroup Forum 93 (2016), no. 3, 491–500. 10.1007/s00233-016-9811-zSearch in Google Scholar

[19] D. Guidetti, B. Karasözen and S. Piskarev, Approximation of abstract differential equations, J. Math. Sci. (N. Y.) 122 (2004), no. 2, 3013–3054. 10.1023/B:JOTH.0000029696.94590.94Search in Google Scholar

[20] B. Jin, B. Li and Z. Zhou, An analysis of the Crank–Nicolson method for subdiffusion, IMA J. Numer. Anal. 38 (2018), no. 1, 518–541. 10.1093/imanum/drx019Search in Google Scholar

[21] B. Jin, B. Li and Z. Zhou, Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math. 138 (2018), no. 1, 101–131. 10.1007/s00211-017-0904-8Search in Google Scholar PubMed PubMed Central

[22] B. Jin, B. Li and Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal. 56 (2018), no. 1, 1–23. 10.1137/16M1089320Search in Google Scholar

[23] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science B.V., Amsterdam, 2006. Search in Google Scholar

[24] B. Kovács, B. Li and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal. 54 (2016), no. 6, 3600–3624. 10.1137/15M1040918Search in Google Scholar

[25] C. Li and M. Li, Hölder regularity for abstract fractional Cauchy problems with order in (0,1), J. Appl. Math. Phys. 6 (2018), 310–319. 10.4236/jamp.2018.61030Search in Google Scholar

[26] M. Li, C. Chen and F.-B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal. 259 (2010), no. 10, 2702–2726. 10.1016/j.jfa.2010.07.007Search in Google Scholar

[27] R. Liu, M. Li, J. Pastor and S. I. Piskarev, On the approximation of fractional resolution families, Differ. Equ. 50 (2014), no. 7, 927–937. 10.1134/S0012266114070088Search in Google Scholar

[28] R. Liu, M. Li and S. Piskarev, Approximation of semilinear fractional Cauchy problem, Comput. Methods Appl. Math. 15 (2015), no. 2, 203–212. 10.1515/cmam-2015-0001Search in Google Scholar

[29] R. Liu, M. Li and S. Piskarev, The order of convergence of difference schemes for fractional equations, Numer. Funct. Anal. Optim. 38 (2017), no. 6, 754–769. 10.1080/01630563.2017.1297825Search in Google Scholar

[30] R. Liu, M. Li and S. I. Piskarev, Stability of difference schemes for fractional equations, Differ. Equ. 51 (2015), no. 7, 904–924. 10.1134/S0012266115070095Search in Google Scholar

[31] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983. 10.1007/978-1-4612-5561-1Search in Google Scholar

[32] S. Piskarev, Differential Equations in Banach Space and Their Approximation (in Russian), Moscow State University Publish House, Moscow, 2005. Search in Google Scholar

[33] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar

[34] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. 10.1007/978-3-0348-8570-6Search in Google Scholar

[35] G. Vainikko, Approximative methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Anal. 2 (1978), no. 6, 647–687. 10.1016/0362-546X(78)90013-5Search in Google Scholar

[36] V. V. Vasil’ev and S. I. Piskarev, Differential equations in Banach spaces. II. Theory of cosine operator functions, J. Math. Sci. (N. Y.) 122 (2004), no. 2, 3055–3174. 10.1023/B:JOTH.0000029697.92324.47Search in Google Scholar

[37] R.-N. Wang, D.-H. Chen and T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations 252 (2012), no. 1, 202–235. 10.1016/j.jde.2011.08.048Search in Google Scholar