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March  2020, 13(3): 609-627. doi: 10.3934/dcdss.2020033

## Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel

 1 Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain 2 African Institute for Mathematical Sciences (AIMS), P.O. Box 608, Limbe Crystal Gardens, South West Region, Cameroon 3 Departamento de Matemática Aplicada Ⅱ, E.E. Aeronáutica e do Espazo, Universidade de Vigo, Campus As Lagoas s/n, 32004 Ourense, Spain

* Corresponding author: Iván Area

Received  April 2018 Revised  May 2018 Published  March 2019

We prove Hölder regularity results for nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel. Existence of weak solutions via approximating solutions is proved. Moreover, Hölder continuity of viscosity solutions is obtained.

Citation: Jean Daniel Djida, Juan J. Nieto, Iván Area. Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 609-627. doi: 10.3934/dcdss.2020033
##### References:
 [1] M. Allen, Hölder regularity for nondivergence nonlocal parabolic equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 110, 29 pp, arXiv: 1610.10073. doi: 10.1007/s00526-018-1367-1. [2] M. Allen, A nondivergence parabolic problem with a fractional time derivative, Differential Integral Equations, 31 (2018), 215-230. [3] M. Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z. [4] I. Area, J. D. Djida, J. Losada and J. J. Nieto, On fractional orthonormal polynomials of a discrete variable, Discrete Dyn. Nat. Soc., 2015 (2015), Article ID 141325, 7 pages. doi: 10.1155/2015/141325. [5] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. [6] A. Bernardis, F. J. Martín-Reyes, P. R. Stinga and J. L. Torrea, Maximum principles, extension problem and inversion for nonlocal one-sided equations, J. Differential Equations, 260 (2016), 6333-6362.  doi: 10.1016/j.jde.2015.12.042. [7] L. Caffarelli, C. H. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Am. Math. Soc., 24 (2011), 849-869.  doi: 10.1090/S0894-0347-2011-00698-X. [8] L. Caffarelli and J. L. Vazquez, Nonlinear porous medium flow with fractional potential pressure, Arch. Rational Mech. Anal., 202 (2011), 537-565.  doi: 10.1007/s00205-011-0420-4. [9] F. Ferrari and I. E. Verbitsky, Radial fractional Laplace operators and hessian inequalities, J. Differential Equations, 253 (2012), 244-272.  doi: 10.1016/j.jde.2012.03.024. [10] R. Herrmann, Fractional Calculus, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2nd edition, 2014. doi: 10.1142/8934. [11] R. Hilfer, Threefold introduction to fractional derivatives, In R. Klages et al. (eds.), editor, Anomalous Transport, (2008), pages 17–77. Wiley-VCH Verlag GmbH & Co. KGaA, 2008. doi: 10.1002/9783527622979.ch2. [12] M. Kassmann, M. Rang and R. W. Schwab, Integro-differential equations with nonlinear directional dependence, Indiana University Mathematics Journal, 63 (2014), 1467-1498.  doi: 10.1512/iumj.2014.63.5394. [13] H. C. Lara and G. Dávila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Differential Equations, 49 (2014), 139-172.  doi: 10.1007/s00526-012-0576-2. [14] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York-London, 1974. [15] S. Samko, A. A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Taylor & Francis, 1993. [16] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion, Adv. Math., 226 (2011), 2020-2039.  doi: 10.1016/j.aim.2010.09.007. [17] L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 843–855, arXiv: 1009.5723. [18] P. R. Stinga and J. L. Torrea, Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation, SIAM J. Math. Anal., 49 (2017), 3893–3924, arXiv: 1511.01945. doi: 10.1137/16M1104317. [19] R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcial. Ekvac., 52 (2009), 1-18.  doi: 10.1619/fesi.52.1.

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##### References:
 [1] M. Allen, Hölder regularity for nondivergence nonlocal parabolic equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 110, 29 pp, arXiv: 1610.10073. doi: 10.1007/s00526-018-1367-1. [2] M. Allen, A nondivergence parabolic problem with a fractional time derivative, Differential Integral Equations, 31 (2018), 215-230. [3] M. Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z. [4] I. Area, J. D. Djida, J. Losada and J. J. Nieto, On fractional orthonormal polynomials of a discrete variable, Discrete Dyn. Nat. Soc., 2015 (2015), Article ID 141325, 7 pages. doi: 10.1155/2015/141325. [5] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. [6] A. Bernardis, F. J. Martín-Reyes, P. R. Stinga and J. L. Torrea, Maximum principles, extension problem and inversion for nonlocal one-sided equations, J. Differential Equations, 260 (2016), 6333-6362.  doi: 10.1016/j.jde.2015.12.042. [7] L. Caffarelli, C. H. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Am. Math. Soc., 24 (2011), 849-869.  doi: 10.1090/S0894-0347-2011-00698-X. [8] L. Caffarelli and J. L. Vazquez, Nonlinear porous medium flow with fractional potential pressure, Arch. Rational Mech. Anal., 202 (2011), 537-565.  doi: 10.1007/s00205-011-0420-4. [9] F. Ferrari and I. E. Verbitsky, Radial fractional Laplace operators and hessian inequalities, J. Differential Equations, 253 (2012), 244-272.  doi: 10.1016/j.jde.2012.03.024. [10] R. Herrmann, Fractional Calculus, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2nd edition, 2014. doi: 10.1142/8934. [11] R. Hilfer, Threefold introduction to fractional derivatives, In R. Klages et al. (eds.), editor, Anomalous Transport, (2008), pages 17–77. Wiley-VCH Verlag GmbH & Co. KGaA, 2008. doi: 10.1002/9783527622979.ch2. [12] M. Kassmann, M. Rang and R. W. Schwab, Integro-differential equations with nonlinear directional dependence, Indiana University Mathematics Journal, 63 (2014), 1467-1498.  doi: 10.1512/iumj.2014.63.5394. [13] H. C. Lara and G. Dávila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Differential Equations, 49 (2014), 139-172.  doi: 10.1007/s00526-012-0576-2. [14] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York-London, 1974. [15] S. Samko, A. A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Taylor & Francis, 1993. [16] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion, Adv. Math., 226 (2011), 2020-2039.  doi: 10.1016/j.aim.2010.09.007. [17] L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 843–855, arXiv: 1009.5723. [18] P. R. Stinga and J. L. Torrea, Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation, SIAM J. Math. Anal., 49 (2017), 3893–3924, arXiv: 1511.01945. doi: 10.1137/16M1104317. [19] R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcial. Ekvac., 52 (2009), 1-18.  doi: 10.1619/fesi.52.1.
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