# American Institute of Mathematical Sciences

August  2019, 24(8): 4031-4053. doi: 10.3934/dcdsb.2019049

## Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation

 1 Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain 2 Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain 3 Departamento de Matemática Aplicada Ⅱ, E.E. Aeronáutica e do Espazo, Universidade de Vigo, Campus As Lagoas s/n, 32004 Vigo, Spain

* Corresponding author: jeandaniel.djida@usc.es

Received  March 2018 Revised  September 2018 Published  February 2019

Fund Project: This work has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016–75140–P, co-financed by the European Community fund FEDER, and Xunta de Galicia, grants GRC 2015–004 and R 2016/022

We study a fractional time porous medium equation with fractional potential pressure. The initial data is assumed to be a bounded function with compact support and fast decay at infinity. We establish existence of weak solutions for which we determine whether the property of compact support is conserved in time depending on some parameters of the problem. Special attention is paid to the property of finite propagation for specific values of the parameters.

Citation: Jean-Daniel Djida, Juan J. Nieto, Iván Area. Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4031-4053. doi: 10.3934/dcdsb.2019049
##### 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. Google Scholar [2] M. Allen, A nondivergence parabolic problem with a fractional time derivative, Differential Integral Equations, 31 (2018), 215–230. Google Scholar [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. Google Scholar [4] M. Allen, L. Caffarelli and A. Vasseur, Porous medium flow with both a fractional potential pressure and fractional time derivative, Chin. Ann. Math. Ser. B, 38 (2017), 45-82. doi: 10.1007/s11401-016-1063-4. Google Scholar [5] I. Area, J. Losada and J. J. Nieto, A note on the fractional logistic equation, Phys. A, 444 (2016), 182-187. doi: 10.1016/j.physa.2015.10.037. Google Scholar [6] D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291. Google Scholar [7] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. doi: 10.2298/TSCI160111018A. Google Scholar [8] D. Baleanu and A. Fernandez, A generalisation of the malgrange-ehrenpreis theorem to find fundamental solutions to fractional PDEs, Electronic Journal of Qualitative Theory of Differential Equations, 2017 (2017), 1-12. doi: 10.14232/ejqtde.2017.1.15. Google Scholar [9] D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444–462. doi: 10.1016/j.cnsns.2017.12.003. Google Scholar [10] 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. Google Scholar [11] P. Biler, C. Imbert and G. Karch, The nonlocal porous medium equation: Barenblatt profiles and other weak solutions, Archive for Rational Mechanics and Analysis, 215 (2015), 497–529. doi: 10.1007/s00205-014-0786-1. Google Scholar [12] P. Biler, G. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Commun. Math. Phys., 294 (2010), 145-168. doi: 10.1007/s00220-009-0855-8. Google Scholar [13] M. Bonforte, A. Figalli and X. Ros-Oton, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains, Comm. Pure Appl. Math., 70 (2017), 1472-1508. doi: 10.1002/cpa.21673. Google Scholar [14] M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018. Google Scholar [15] C. Bucur, Some nonlocal operators and effects due to nonlocality, preprint, arXiv: 1705.00953Google Scholar [16] C. Bucur and F. Ferrari, An extension problem for the fractional derivative defined by Marchaud, Fract. Calc. Appl. Anal., 19 (2016), 867-887. doi: 10.1515/fca-2016-0047. Google Scholar [17] L. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, Journal Europ. Math. Society., 15 (2013), 1701-1746. doi: 10.4171/JEMS/401. Google Scholar [18] L. Caffarelli and J. L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 1393-1404. doi: 10.3934/dcds.2011.29.1393. Google Scholar [19] L. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal., 202 (2011), 537-565. doi: 10.1007/s00205-011-0420-4. Google Scholar [20] M. Caputo, Diffusion of fluids in porous media with memory, Geothermics, 28 (1999), 113-130. doi: 10.1016/S0375-6505(98)00047-9. Google Scholar [21] M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr Fract Differ Appl, 1 (2015), 73-85. Google Scholar [22] J. A. Carrillo, Y. Huang, M. C. Santos and J. L. Vázquez, Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure, J. Differ. Equations, 258 (2015), 736-763. doi: 10.1016/j.jde.2014.10.003. Google Scholar [23] J. D. Djida, A. Atangana and I. Area, Numerical computation of a fractional derivative with non-local and non-singular kernel, Math. Model. Nat. Phenom., 12 (2017), 4-13. doi: 10.1051/mmnp/201712302. Google Scholar [24] J. D. Djida, J. J. Nieto and I. Area, Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel, Discrete Continuous Dyn. Syst. Ser. S (to appear), 2018.Google Scholar [25] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Masson, Paris, 1987. Google Scholar [26] A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017. Google Scholar [27] A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284. doi: 10.1002/cpa.21408. Google Scholar [28] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [29] S. Dipierro, E. Valdinoci and V. Vespri, Decay estimates for evolutionary equations with fractional time diffusion, preprint, arXiv: 1707.08278v1.Google Scholar [30] F. Ferrari, Weyl and Marchaud derivatives: A forgotten history, Mathematics, 6 (2018), p6. doi: 10.3390/math6010006. Google Scholar [31] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Berlin: Springer, reprint of the 1998 ed. edition, 2001. Google Scholar [32] J. Kemppainen, J. Siljander and R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, Journal of Differential Equations, 263 (2017), 149-201. doi: 10.1016/j.jde.2017.02.030. Google Scholar [33] P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Communications on Pure and Applied Analysis, 13 (2014), 2543-2557. doi: 10.3934/cpaa.2014.13.2543. Google Scholar [34] A. N. Kochubei, Fractional-order diffusion, Differ. Equations, 26 (1990), 485-492. Google Scholar [35] L. Liu, T. Caraballo and P. E. Kloeden, Long time behavior of stochastic parabolic problems with white noise in materials with thermal memory, Revista Matemática Complutense, 30 (2018), 687-717. doi: 10.1007/s13163-017-0238-1. Google Scholar [36] F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192. Google Scholar [37] F. Mainardi and G. Pagnini, The Wright functions as solutions of the time-fractional diffusion equation, Applied Mathematics and Computation, 141 (2003), 51-62. doi: 10.1016/S0096-3003(02)00320-X. Google Scholar [38] A. Marchaud, Sur Les Dérivées et Sur les Différences des Fonctions De Variables Réelles, PhD thesis, Faculté des Sciences de Paris, 1927. Google Scholar [39] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. Google Scholar [40] G. M. Mophou and G. M. N'Guérékata, On a class of fractional differential equations in a Sobolev space, Applicable Analysis, 91 (2012), 15-34. doi: 10.1080/00036811.2010.534730. Google Scholar [41] M. K. Saad, A. Atangana and D. Baleanu, New fractional derivatives with non-singular kernel applied to the Burgers equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284. Google Scholar [42] S. Samko, A. A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Taylor & Francis, 1993. Google Scholar [43] J. Simon, Compact sets in the space $L^p(0, t; b)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65–96. doi: 10.1007/BF01762360. Google Scholar [44] D. Stan, F. del Teso and J. L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure, Comptes Rendus Acad. Sci., 352 (2014), 123-128. doi: 10.1016/j.crma.2013.12.003. Google Scholar [45] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Google Scholar [46] J. L. Vázquez, The Porous Medium Equation, Oxford University Press, 2007. Google Scholar [47] J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, Nonlinear Partial Differential Equations, 271–298, Abel Symp., 7, Springer, Heidelberg, 2012. Available online at http://www.uam.es/personal_pdi/ciencias/jvazquez/JLV ABEL-2010.pdf. doi: 10.1007/978-3-642-25361-4_15. Google Scholar

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. Google Scholar [2] M. Allen, A nondivergence parabolic problem with a fractional time derivative, Differential Integral Equations, 31 (2018), 215–230. Google Scholar [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. Google Scholar [4] M. Allen, L. Caffarelli and A. Vasseur, Porous medium flow with both a fractional potential pressure and fractional time derivative, Chin. Ann. Math. Ser. B, 38 (2017), 45-82. doi: 10.1007/s11401-016-1063-4. Google Scholar [5] I. Area, J. Losada and J. J. Nieto, A note on the fractional logistic equation, Phys. A, 444 (2016), 182-187. doi: 10.1016/j.physa.2015.10.037. Google Scholar [6] D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291. Google Scholar [7] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. doi: 10.2298/TSCI160111018A. Google Scholar [8] D. Baleanu and A. Fernandez, A generalisation of the malgrange-ehrenpreis theorem to find fundamental solutions to fractional PDEs, Electronic Journal of Qualitative Theory of Differential Equations, 2017 (2017), 1-12. doi: 10.14232/ejqtde.2017.1.15. Google Scholar [9] D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444–462. doi: 10.1016/j.cnsns.2017.12.003. Google Scholar [10] 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. Google Scholar [11] P. Biler, C. Imbert and G. Karch, The nonlocal porous medium equation: Barenblatt profiles and other weak solutions, Archive for Rational Mechanics and Analysis, 215 (2015), 497–529. doi: 10.1007/s00205-014-0786-1. Google Scholar [12] P. Biler, G. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Commun. Math. Phys., 294 (2010), 145-168. doi: 10.1007/s00220-009-0855-8. Google Scholar [13] M. Bonforte, A. Figalli and X. Ros-Oton, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains, Comm. Pure Appl. Math., 70 (2017), 1472-1508. doi: 10.1002/cpa.21673. Google Scholar [14] M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018. Google Scholar [15] C. Bucur, Some nonlocal operators and effects due to nonlocality, preprint, arXiv: 1705.00953Google Scholar [16] C. Bucur and F. Ferrari, An extension problem for the fractional derivative defined by Marchaud, Fract. Calc. Appl. Anal., 19 (2016), 867-887. doi: 10.1515/fca-2016-0047. Google Scholar [17] L. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, Journal Europ. Math. Society., 15 (2013), 1701-1746. doi: 10.4171/JEMS/401. Google Scholar [18] L. Caffarelli and J. L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 1393-1404. doi: 10.3934/dcds.2011.29.1393. Google Scholar [19] L. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal., 202 (2011), 537-565. doi: 10.1007/s00205-011-0420-4. Google Scholar [20] M. Caputo, Diffusion of fluids in porous media with memory, Geothermics, 28 (1999), 113-130. doi: 10.1016/S0375-6505(98)00047-9. Google Scholar [21] M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr Fract Differ Appl, 1 (2015), 73-85. Google Scholar [22] J. A. Carrillo, Y. Huang, M. C. Santos and J. L. Vázquez, Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure, J. Differ. Equations, 258 (2015), 736-763. doi: 10.1016/j.jde.2014.10.003. Google Scholar [23] J. D. Djida, A. Atangana and I. Area, Numerical computation of a fractional derivative with non-local and non-singular kernel, Math. Model. Nat. Phenom., 12 (2017), 4-13. doi: 10.1051/mmnp/201712302. Google Scholar [24] J. D. Djida, J. J. Nieto and I. Area, Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel, Discrete Continuous Dyn. Syst. Ser. S (to appear), 2018.Google Scholar [25] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Masson, Paris, 1987. Google Scholar [26] A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017. Google Scholar [27] A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284. doi: 10.1002/cpa.21408. Google Scholar [28] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [29] S. Dipierro, E. Valdinoci and V. Vespri, Decay estimates for evolutionary equations with fractional time diffusion, preprint, arXiv: 1707.08278v1.Google Scholar [30] F. Ferrari, Weyl and Marchaud derivatives: A forgotten history, Mathematics, 6 (2018), p6. doi: 10.3390/math6010006. Google Scholar [31] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Berlin: Springer, reprint of the 1998 ed. edition, 2001. Google Scholar [32] J. Kemppainen, J. Siljander and R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, Journal of Differential Equations, 263 (2017), 149-201. doi: 10.1016/j.jde.2017.02.030. Google Scholar [33] P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Communications on Pure and Applied Analysis, 13 (2014), 2543-2557. doi: 10.3934/cpaa.2014.13.2543. Google Scholar [34] A. N. Kochubei, Fractional-order diffusion, Differ. Equations, 26 (1990), 485-492. Google Scholar [35] L. Liu, T. Caraballo and P. E. Kloeden, Long time behavior of stochastic parabolic problems with white noise in materials with thermal memory, Revista Matemática Complutense, 30 (2018), 687-717. doi: 10.1007/s13163-017-0238-1. Google Scholar [36] F. Mainardi, Y. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192. Google Scholar [37] F. Mainardi and G. Pagnini, The Wright functions as solutions of the time-fractional diffusion equation, Applied Mathematics and Computation, 141 (2003), 51-62. doi: 10.1016/S0096-3003(02)00320-X. Google Scholar [38] A. Marchaud, Sur Les Dérivées et Sur les Différences des Fonctions De Variables Réelles, PhD thesis, Faculté des Sciences de Paris, 1927. Google Scholar [39] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. Google Scholar [40] G. M. Mophou and G. M. N'Guérékata, On a class of fractional differential equations in a Sobolev space, Applicable Analysis, 91 (2012), 15-34. doi: 10.1080/00036811.2010.534730. Google Scholar [41] M. K. Saad, A. Atangana and D. Baleanu, New fractional derivatives with non-singular kernel applied to the Burgers equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284. Google Scholar [42] S. Samko, A. A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Taylor & Francis, 1993. Google Scholar [43] J. Simon, Compact sets in the space $L^p(0, t; b)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65–96. doi: 10.1007/BF01762360. Google Scholar [44] D. Stan, F. del Teso and J. L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure, Comptes Rendus Acad. Sci., 352 (2014), 123-128. doi: 10.1016/j.crma.2013.12.003. Google Scholar [45] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Google Scholar [46] J. L. Vázquez, The Porous Medium Equation, Oxford University Press, 2007. Google Scholar [47] J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, Nonlinear Partial Differential Equations, 271–298, Abel Symp., 7, Springer, Heidelberg, 2012. Available online at http://www.uam.es/personal_pdi/ciencias/jvazquez/JLV ABEL-2010.pdf. doi: 10.1007/978-3-642-25361-4_15. Google Scholar
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