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An \(L_{q}(L_{p})\)-theory for the time-fractional evolution equations with variable coefficients. (English) Zbl 1361.35196

Summary: We introduce an \(L_q(L_p)\)-theory for the semilinear fractional equations of the type[(0.1)] \(\partial_t^\alpha u(t, x) = a^{i j}(t, x) u_{x^i x^j}(t, x) + f(t, x, u), t > 0, x \in \mathbb{R}^d .\) Here, \(\alpha \in(0, 2)\), \(p, q > 1\), and \(\partial_t^\alpha\) is the Caupto fractional derivative of order \(\alpha\). Uniqueness, existence, and \(L_q(L_p)\)-estimates of solutions are obtained. The leading coefficients \(a^{i j}(t, x)\) are assumed to be piecewise continuous in \(t\) and uniformly continuous in \(x\). In particular, \(a^{i j}(t, x)\) are allowed to be discontinuous with respect to the time variable. Our approach is based on classical tools in PDE theories such as the Marcinkiewicz interpolation theorem, the Calderon-Zygmund theorem, and perturbation arguments.

MSC:

35R11 Fractional partial differential equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
26A33 Fractional derivatives and integrals
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[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, vol. 55 (1972), Courier Dover Publications · Zbl 0543.33001
[2] Andrews, G. E.; Askey, R.; Roy, R., Special Functions, vol. 71 (1999), Cambridge University Press · Zbl 0920.33001
[3] Bagley, R. L.; Torvik, P., A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheology, 27, 3, 201-210 (1983), (1978-present) · Zbl 0515.76012
[4] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., Application of a fractional advection-dispersion equation, Water Resour. Res., 36, 6, 1403-1412 (2000)
[5] Braaksma, B. L.J., Asymptotic expansions and analytic continuations for a class of barnes-integrals, Compos. Math., 15, 239-341 (1936) · Zbl 0129.28604
[6] Caponetto, R., Fractional Order Systems: Modeling and Control Applications, vol. 72 (2010), World Scientific
[7] Clément, P.; Londen, S.-O.; Simonett, G., Quasilinear evolutionary equations and continuous interpolation spaces, J. Differential Equations, 196, 2, 418-447 (2004) · Zbl 1058.35136
[8] Clément, P.; Prüss, J., Global existence for a semilinear parabolic volterra equation, Math. Z., 209, 1, 17-26 (1992) · Zbl 0724.45012
[9] Da Prato, G.; Iannelli, M., Existence and regularity for a class of integrodifferential equations of parabolic type, J. Math. Anal. Appl., 112, 1, 36-55 (1985) · Zbl 0583.45009
[10] Djrbashian, M. M., Harmonic Analysis and Boundary Value Problems in the Complex Domain, vol. 65 (1993), Springer · Zbl 0816.30023
[11] Eidelman, S. D.; Ivasyshen, S. D.; Kochubei, A. N., Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, vol. 152 (2004), Springer · Zbl 1129.35427
[12] Eidelman, S. D.; Kochubei, A. N., Cauchy problem for fractional diffusion equations, J. Differential Equations, 199, 2, 211-255 (2004) · Zbl 1129.35427
[13] Engheia, N., On the role of fractional calculus in electromagnetic theory, IEEE Antennas and Propagation Mag., 39, 4, 35-46 (1997)
[14] Glöckle, W. G.; Nonnenmacher, T. F., A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68, 1, 46-53 (1995)
[15] Hilfer, R.; Butzer, P.; Westphal, U.; Douglas, J.; Schneider, W.; Zaslavsky, G.; Nonnemacher, T.; Blumen, A.; West, B., Applications of Fractional Calculus in Physics, vol. 5 (2000), World Scientific
[16] Kilbas, A. A., H-Transforms: Theory and Applications (2004), CRC Press · Zbl 1056.44001
[17] Kim, I.; Kim, K.-H.; Lim, S., Parabolic BMO estimates for pseudo-differential operators of arbitrary order (2014), arXiv preprint
[18] Kochubei, A. N., Asymptotic properties of solutions of the fractional diffusion-wave equation, Fract. Calc. Appl. Anal., 17, 3, 881-896 (2014) · Zbl 1323.35197
[19] Krylov, N. V., An analytic approach to SPDEs, (Stochastic Partial Differential Equations: Six Perspectives. Stochastic Partial Differential Equations: Six Perspectives, Math. Surveys Monogr., vol. 64 (1999)), 185-242 · Zbl 0933.60073
[20] Krylov, N. V., On the Calderón-Zygmund theorem with applications to parabolic equations, Algebra i Analiz, 13, 4, 1-25 (2001) · Zbl 1011.35033
[21] Kunstmann, P. C.; Weis, L., Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \)-functional calculus, (Functional Analytic Methods for Evolution Equations (2004), Springer), 65-311 · Zbl 1097.47041
[22] Langlands, T.; Henry, B.; Wearne, S., Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions, J. Math. Biol., 59, 6, 761-808 (2009) · Zbl 1232.92037
[23] Mainardi, F., Fractional diffusive waves in viscoelastic solids, (Nonlinear Waves in Solids (1995), Fairfield), 93-97
[24] Meerschaert, M. M.; Sikorskii, A., Stochastic Models for Fractional Calculus, vol. 43 (2011), Walter de Gruyter
[25] Metzler, R.; Barkai, E.; Klafter, J., Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82, 18, 3563 (1999)
[26] Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Phys. A, 278, 1, 107-125 (2000)
[27] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1, 1-77 (2000) · Zbl 0984.82032
[28] Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen., 37, 31, Article R161 pp. (2004) · Zbl 1075.82018
[29] Metzler, R.; Schick, W.; Kilian, H.-G.; Nonnenmacher, T. F., Relaxation in filled polymers: a fractional calculus approach, J. Chem. Phys., 103, 16, 7180-7186 (1995)
[30] Ortigueira, M. D., Fractional Calculus for Scientists and Engineers, vol. 84 (2011), Springer · Zbl 1251.26005
[31] Podlubny, I., Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 (1998), Academic Press
[32] Podlubny, I., Fractional-order systems and \(P I^\lambda D^\mu \)-controllers, IEEE Trans. Automat. Control, 44, 1, 208-214 (1999) · Zbl 1056.93542
[33] Prüss, J., Quasilinear parabolic Volterra equations in spaces of integrable functions, (de Pagter, B.; Clément, Ph.; Mitidieri, E., Semigroup Theory and Evolution Equations. Semigroup Theory and Evolution Equations, Lect. Notes Pure Appl. Math., vol. 135 (1991)), 401-420 · Zbl 0742.45007
[34] Prüss, J., Evolutionary Integral Equations and Applications (2012), Springer · Zbl 1258.45008
[35] Pskhu, A. V., The fundamental solution of a diffusion-wave equation of fractional order, Izv. Math., 73, 2, 351 (2009) · Zbl 1172.26001
[36] Raberto, M.; Scalas, E.; Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Phys. A, 314, 1, 749-755 (2002) · Zbl 1001.91033
[37] Sabatier, J.; Agrawal, O. P.; Machado, J. T., Advances in Fractional Calculus (2007), Springer · Zbl 1116.00014
[38] Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 1, 426-447 (2011) · Zbl 1219.35367
[39] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, vol. 1993 (1993) · Zbl 0818.26003
[40] Scalas, E.; Gorenflo, R.; Mainardi, F., Fractional calculus and continuous-time finance, Phys. A, 284, 1, 376-384 (2000)
[41] Schumer, R.; Benson, D. A.; Meerschaert, M. M.; Wheatcraft, S. W., Eulerian derivation of the fractional advection-dispersion equation, J. Contam. Hydrol., 48, 1, 69-88 (2001)
[42] Stein, E. M.; Weiss, G. L., Introduction to Fourier Analysis on Euclidean Spaces, vol. 1 (1971), Princeton University Press · Zbl 0232.42007
[43] Tarasov, V. E., Electromagnetic fields on fractals, Modern Phys. Lett. A, 21, 20, 1587-1600 (2006) · Zbl 1097.78003
[44] von Wolfersdorf, L., On identification of memory kernels in linear theory of heat conduction, Math. Methods Appl. Sci., 17, 12, 919-932 (1994) · Zbl 0811.45004
[45] Ye, H.; Gao, J.; Ding, Y., A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328, 2, 1075-1081 (2007) · Zbl 1120.26003
[46] Zacher, R., Maximal regularity of type \(L_p\) for abstract parabolic Volterra equations, J. Evol. Equ., 5, 1, 79-103 (2005) · Zbl 1104.45008
[47] Zacher, R., Quasilinear parabolic integro-differential equations with nonlinear boundary conditions, Differential Integral Equations, 19, 10, 1129-1156 (2006) · Zbl 1212.45015
[48] Zacher, R., Weak solutions of abstract evolutionary integro-differential equations in hilbert spaces, Funkcial. Ekvac., 52, 1, 1-18 (2009) · Zbl 1171.45003
[49] Zacher, R., Quasiliner Parabolic Problems with Nonlinear Boundary Conditions (2003), Martin-Luther-Universität Halle, Thesis · Zbl 1366.35096
[50] Zaslavsky, G. M., Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371, 6, 461-580 (2002) · Zbl 0999.82053
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