# American Institute of Mathematical Sciences

October  2011, 29(4): 1393-1404. doi: 10.3934/dcds.2011.29.1393

## Asymptotic behaviour of a porous medium equation with fractional diffusion

 1 Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1082 2 Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid

Received  March 2010 Revised  August 2010 Published  December 2010

We consider a porous medium equation with a nonlocal diffusion effect given by an inverse fractional Laplacian operator. The equation is posed in the whole space $\mathbb{R}^n$. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy estimates. Here we establish the large-time behaviour. We first find selfsimilar nonnegative solutions by solving an elliptic obstacle problem for the pair pressure-density involving the Laplacian, obtaining what we call obstacle Barenblatt solutions. The theory for elliptic fractional problems with obstacles has been recently established. We then use entropy methods to show that the asymptotic behavior of general finite-mass solutions is described after renormalization by these special solutions, which represent a surprising variation of the Barenblatt profiles of the standard porous medium model.
Citation: Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
##### References:
 [1] I. Athanasopoulos, L. A. Caffarelli and S. Salsa, The structure of the free boundary for lower dimensional obstacle problems, Amer. J. Math., 130 (2008), 485-498. doi: 10.1353/ajm.2008.0016. [2] G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., 16 (1952), 679-698. [3] P. Biler, C. Imbert and G. Karch, Fractal porous medium equation, preprint. [4] P. Biler, G. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Comm. Math. Phys., 294 (2010), 145-168. doi: 10.1007/s00220-009-0855-8. [5] L. A. Caffarelli, The obstacle problem revisited, The Journal of Fourier Analysis and Applications, 4 (1998), 383-402. doi: 10.1007/BF02498216. [6] L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119. [7] L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6. [8] L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqns., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [9] L. A. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, in preparation. [10] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [11] L. A. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure, arXiv:1001.0410. [12] J. A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), 113-142. doi: 10.1512/iumj.2000.49.1756. [13] J. A. Carrillo, A. Jngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82. doi: 10.1007/s006050170032. [14] M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., 81 (2002), 847-875. doi: 10.1016/S0021-7824(02)01266-7. [15] A. Friedman, "Variational Principles and Free Boundary Problems," Wiley, New York, 1982. [16] A. Friedman and S. Kamin, The asymptotic behavior of gas in an $N$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563. [17] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer, Berlin, 2001. (reprint of the 1998 edition) [18] A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation, Phil. Mag., 26 (1972), 65-72. doi: 10.1080/14786437208221020. [19] N. S. Landkof, "Foundations Of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180 (translated from the Russian by A. P. Doohovskoy), Springer, New York, 1972. [20] L. E. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706. [21] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. [22] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. [23] J. L. Vázquez, "Smoothing And Decay Estimates For Nonlinear Diffusion Equations. Equations Of Porous Medium Type," Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006.

show all references

##### References:
 [1] I. Athanasopoulos, L. A. Caffarelli and S. Salsa, The structure of the free boundary for lower dimensional obstacle problems, Amer. J. Math., 130 (2008), 485-498. doi: 10.1353/ajm.2008.0016. [2] G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., 16 (1952), 679-698. [3] P. Biler, C. Imbert and G. Karch, Fractal porous medium equation, preprint. [4] P. Biler, G. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Comm. Math. Phys., 294 (2010), 145-168. doi: 10.1007/s00220-009-0855-8. [5] L. A. Caffarelli, The obstacle problem revisited, The Journal of Fourier Analysis and Applications, 4 (1998), 383-402. doi: 10.1007/BF02498216. [6] L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119. [7] L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6. [8] L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqns., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [9] L. A. Caffarelli, F. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, in preparation. [10] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. [11] L. A. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure, arXiv:1001.0410. [12] J. A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), 113-142. doi: 10.1512/iumj.2000.49.1756. [13] J. A. Carrillo, A. Jngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82. doi: 10.1007/s006050170032. [14] M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., 81 (2002), 847-875. doi: 10.1016/S0021-7824(02)01266-7. [15] A. Friedman, "Variational Principles and Free Boundary Problems," Wiley, New York, 1982. [16] A. Friedman and S. Kamin, The asymptotic behavior of gas in an $N$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563. [17] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer, Berlin, 2001. (reprint of the 1998 edition) [18] A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation, Phil. Mag., 26 (1972), 65-72. doi: 10.1080/14786437208221020. [19] N. S. Landkof, "Foundations Of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180 (translated from the Russian by A. P. Doohovskoy), Springer, New York, 1972. [20] L. E. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706. [21] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. [22] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. [23] J. L. Vázquez, "Smoothing And Decay Estimates For Nonlinear Diffusion Equations. Equations Of Porous Medium Type," Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006.
 [1] Sofía Nieto, Guillermo Reyes. Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1123-1139. doi: 10.3934/cpaa.2013.12.1123 [2] Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5927-5962. doi: 10.3934/dcds.2015.35.5927 [3] Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011 [4] Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks and Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337 [5] Matteo Bonforte, Yannick Sire, Juan Luis Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725 [6] Nikolaos Roidos, Yuanzhen Shao. Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation. Evolution Equations and Control Theory, 2022, 11 (3) : 793-825. doi: 10.3934/eect.2021026 [7] Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595 [8] Goro Akagi. Energy solutions of the Cauchy-Neumann problem for porous medium equations. Conference Publications, 2009, 2009 (Special) : 1-10. doi: 10.3934/proc.2009.2009.1 [9] María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations and Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001 [10] Ahmad Z. Fino, Mokhtar Kirane. The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3625-3650. doi: 10.3934/cpaa.2020160 [11] Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783 [12] Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure and Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623 [13] Xinfu Chen, Jong-Shenq Guo, Bei Hu. Dead-core rates for the porous medium equation with a strong absorption. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1761-1774. doi: 10.3934/dcdsb.2012.17.1761 [14] Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure and Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183 [15] Zhilei Liang. On the critical exponents for porous medium equation with a localized reaction in high dimensions. Communications on Pure and Applied Analysis, 2012, 11 (2) : 649-658. doi: 10.3934/cpaa.2012.11.649 [16] Edoardo Mainini. On the signed porous medium flow. Networks and Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525 [17] Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021 [18] Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure and Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 [19] Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905 [20] Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111

2020 Impact Factor: 1.392