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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 & Continuous Dynamical Systems - S, 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. AllenL. 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. BernardisF. J. Martín-ReyesP. 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. CaffarelliC. 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. KassmannM. 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.

show all references

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. AllenL. 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. BernardisF. J. Martín-ReyesP. 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. CaffarelliC. 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. KassmannM. 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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