Abstract fractional Cauchy problems with almost sectorial operators. (English) Zbl 1238.34015

In the first part of this paper, after a brief overview of the construction of functional calculus about almost sectorial operators, the authors state some results about the analytic semigroups of growth order \(1+\gamma\) and summarize some properties on Caputo fractional derivative and two special functions. Next, a pair of families of operators is constructed and some properties for these families are given. This enables the authors to introduce a concept of solution, which is used to analyze the existence of mild solutions and classical solutions to the Cauchy problem. The corresponding semilinear problem is studied in the next section of this paper. First, the existence of mild solutions is investigated, and then the existence of classical solutions. Some examples are provided in the final section of this paper. They cover the cases of systems of fractional partial differential equations, fractional initial-boundary value problems, and fractional Cauchy problems.


34A08 Fractional ordinary differential equations
35K90 Abstract parabolic equations
47A60 Functional calculus for linear operators
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
Full Text: DOI


[1] Anh, Vo V.; Leonenko, N. N., Spectral analysis of fractional kinetic equations with random data, J. Stat. Phys., 104, 1349-1387 (2001) · Zbl 1034.82044
[2] Arrieta, J. M., Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case, Trans. Amer. Math. Soc., 347, 9, 3503-3531 (1995) · Zbl 0856.35095
[3] Arrieta, J. M.; Carvalho, A.; Lozada-Cruz, G., Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations, 231, 551-597 (2006) · Zbl 1110.35028
[4] Arrieta, J. M.; Carvalho, A.; Lozada-Cruz, G., Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations, 247, 174-202 (2009) · Zbl 1172.35033
[5] Arrieta, J. M.; Carvalho, A.; Lozada-Cruz, G., Dynamics in dumbbell domains III. Continuity of attractors, J. Differential Equations, 247, 225-259 (2009) · Zbl 1172.35034
[6] Carvalho, A. N.; Dlotko, T.; Nescimento, M. J.D., Non-autonomous semilinear evolution equations with almost sectorial operators, J. Evol. Equ., 8, 631-659 (2008) · Zbl 1180.35319
[7] del-Castillo-Negrete, D.; Carreras, B. A.; Lynch, V., Front dynamics in reaction-diffusion systems with Levy flights: A fractional diffusion approach, Phys. Rev. Lett., 91, 1, 018302 (2003)
[8] Cowling, M.; Doust, I.; McIntosh, A.; Yagi, A., Banach space operators with a bounded \(H^\infty\) calculus, J. Aust. Math. Soc. Sect. A, 60, 51-89 (1996) · Zbl 0853.47010
[9] Dancer, E. N.; Daners, D., Domain perturbation of elliptic equations subject to Robin boundary conditions, J. Differential Equations, 74, 86-132 (1997) · Zbl 0886.35063
[10] deLaubenfels, R., Existence Families, Functional Calculi and Evolution Equations (1994), Springer-Verlag: Springer-Verlag Berlin · Zbl 0811.47034
[11] Diethelm, K.; Freed, A. D., On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, (Keil, F.; Mackens, W.; Voss, H.; Werther, J., Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties (1999), Springer-Verlag: Springer-Verlag Heidelberg), 217-224
[12] Ducrot, A.; Magal, P.; Prevost, K., Integrated semigroups and parabolic equations. Part I: linear perturbation of almost sectorial operators, J. Evol. Equ., 10, 263-291 (2010) · Zbl 1239.35083
[13] Duncan, T. E.; Hu, Y.; Pasik-Duncan, B., Stochastic calculus for fractional Brownian motion, I. Theory, SIAM J. Control Optim., 38, 582-612 (2000) · Zbl 0947.60061
[14] Dzhrbashyan, M. M.; Nersessyan, A. B., Fractional derivatives and Cauchy problem for differential equations of fractional order, Izv. AN Arm. SSR. Mat., 3, 3-29 (1968), (in Russian) · Zbl 0165.40801
[15] Eidelman, Samuil D.; Kochubei, Anatoly N., Cauchy problem for fractional diffusion equations, J. Differential Equations, 199, 2, 211-255 (2004) · Zbl 1068.35037
[16] Gadylʼshin, R. R., On the eigenvalues of a dumb-bell with a thin handle, Izv. Math., 69, 2, 265-329 (2005) · Zbl 1075.35023
[17] Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Springer: Springer Berlin · Zbl 0456.35001
[18] Hernandez, E.; OʼRegan, D.; Balachandran, E., On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., 73, 3462-3471 (2010) · Zbl 1229.34004
[19] Hilfer, H., Applications of Fractional Calculus in Physics (2000), World Scientific Publ. Co.: World Scientific Publ. Co. Singapore · Zbl 0998.26002
[20] Jaradat, O. K.; Al-Omari, A.; Momani, S., Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal., 69, 3153-3159 (2008) · Zbl 1160.34300
[21] Jimbo, S., The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary conditions, J. Differential Equations, 77, 322-350 (1989) · Zbl 0703.35138
[22] Hu, Y.; Kallianpur, G., Schröodinger equations with fractional Laplacians, Appl. Math. Optim., 42, 281-290 (2000) · Zbl 1002.60048
[23] Kilbas, A. A.; Srivastava, Hari M.; Trujillo, J. Juan, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., vol. 204 (2006), Elsevier Science B.V.: Elsevier Science B.V. Amsterdam · Zbl 1092.45003
[24] Kirane, M.; Laskri, Y.; Tatar, N.-e., Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives, J. Math. Anal. Appl., 312, 488-501 (2005) · Zbl 1135.35350
[25] Kirane, M.; Malik, S. A., The profile of blowing-up solutions to a nonlinear system of fractional differential equations, Nonlinear Anal., 73, 3723-3736 (2010) · Zbl 1205.26013
[26] Kirane, M.; Tatar, N.-e., Exponential growth for a fractionally damped wave equation, Z. Anal. Anwend., 22, 167-177 (2003) · Zbl 1029.35177
[27] Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems (1995), Birkhäuser: Birkhäuser Basel · Zbl 0816.35001
[28] Mainardi, F., Fractional calculus: Some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag Wien), 291-348 · Zbl 0917.73004
[29] Markus, H., The Functional Calculus for Sectorial Operators, Oper. Theory Adv. Appl., vol. 69 (2006), Birkhäuser-Verlag: Birkhäuser-Verlag Basel · Zbl 1101.47010
[30] Maslowski, B.; Nualart, D., Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202, 277-305 (2003) · Zbl 1027.60060
[31] McIntosh, A., Operators which have an \(H^\infty\) functional calculus, (Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14 (1986)), 210-231 · Zbl 0634.47016
[32] Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F., Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys., 103, 7180-7186 (1995)
[33] Metzler, R.; Klafter, J., The random walkʼs guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032
[34] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Differential Equations (1993), John Wiley: John Wiley New York · Zbl 0789.26002
[35] van Neerven, J. M.A. M.; Straub, B., On the existence and growth of mild solutions of the abstract Cauchy problem for operators with polynomially bounded resolvent, Houston J. Math., 24, 137-171 (1998) · Zbl 0966.34050
[36] Nualart, D.; Vuillermot, P., Variational solutions for partial differential equations driven by fractional noise, J. Funct. Anal., 232, 390-454 (2006) · Zbl 1089.35097
[37] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023
[38] Periago, F.; Straub, B., A functional calculus for almost sectorial operators and applications to abstract evolution equations, J. Evol. Equ., 2, 41-68 (2002) · Zbl 1005.47015
[39] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[40] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003
[41] Toropova, S. P., Relation of semigroups with singularity to integrated semigroups, Russian Math., 47, 66-74 (2003) · Zbl 1091.47035
[42] von Wahl, W., Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., 11, 231-258 (1972) · Zbl 0251.35052
[43] Xiao, T. J.; Liang, J., The Cauchy Problem for Higher Order Abstract Differential Equations, Lecture Notes in Math., vol. 1701 (1998), Springer: Springer Berlin, New York · Zbl 0915.34002
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