# American Institute of Mathematical Sciences

December  2015, 35(12): 5725-5767. doi: 10.3934/dcds.2015.35.5725

## Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains

 1 Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain, Spain 2 Université Aix-Marseille, I2M, Centre de Mathématique et Informatique, Technopôle de Chateau-Giombert, Marseille, France

Received  April 2014 Published  May 2015

We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables solutions of the friendly giant type. As a by-product, we obtain an existence and uniqueness result for semilinear elliptic non local equations with sub-linear nonlinearities. The Appendix contains a review of the theory of fractional Sobolev spaces and of the interpolation theory that are used in the rest of the paper.
Citation: 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
##### References:
 [1] R. A. Adams and J. F. Fournier, Sobolev spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. [2] D. G. Aronson and L. A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains, J. Diff. Equations, 39 (1981), 378-412. doi: 10.1016/0022-0396(81)90065-6. [3] N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Tech. Report of Univ. of Kansas, 14 (1955), 77-94. [4] I. Athanasopoulos and L. A. Caffarelli, Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315. doi: 10.1016/j.aim.2009.11.010. [5] P. Bénilan and M. G. Crandall, The continuous dependence on $\varphi$ of solutions of $u_t - \Delta \varphi (u) = 0$, Indiana Univ. Math. J., 30 (1981), 161-177. doi: 10.1512/iumj.1981.30.30014. [6] R. M. Blumenthal and R. K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math., 9 (1959), 399-408. doi: 10.2140/pjm.1959.9.399. [7] M. Bonforte, G. Grillo and J. L. Vázquez, Fast diffusion flow on manifolds of nonpositive curvature, J. Evol. Eq., 8 (2008), 99-128. doi: 10.1007/s00028-007-0345-4. [8] M. Bonforte, G. Grillo and J. L. Vázquez, Behaviour near extinction for the fast diffusion equation on bounded domains, J. Math. Pures Appl., 97 (2012), 1-38. doi: 10.1016/j.matpur.2011.03.002. [9] M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Advances in Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018. [10] M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Arch. Ration. Mech. Anal., appeared online. doi: 10.1007/s00205-015-0861-2. [11] M. Bonforte and J. L. Vázquez, Nonlinear Degenerate Diffusion Equations on bounded domains with Restricted Fractional Laplacian, in preparation, 2014. [12] H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), Acad. Press, 1971, 101-156. [13] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland, 1973. [14] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. [15] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eq., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [16] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilinear equations, Comm. Partial Diff. Eq., 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. [17] Z. Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal., 226 (2005), 90-113. doi: 10.1016/j.jfa.2005.05.004. [18] A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034. [19] B. Dahlberg and C. Kenig, Nonnegative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Amer. Math. Soc., 1 (1988), 401-412. doi: 10.1090/S0894-0347-1988-0928264-9. [20] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990. [21] M. Crandall and M. Pierre, Regularizing Effects for $u_t=A\varphi(u)$ in $\mathbbL^1$, J. Funct. Anal., 45 (1982), 194-212. doi: 10.1016/0022-1236(82)90018-0. [22] E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623721. [23] E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal., 59 (1984), 335-395. doi: 10.1016/0022-1236(84)90076-4. [24] E. Gagliardo, Proprietá di alcune classi di funzioni in piú variabili, Ric. Mat., 7 (1958), 102-137. [25] G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators, Adv. in Math., 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018. [26] T. Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist., 22 (2002), 419-441. [27] T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. [28] N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. [29] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374. doi: 10.2307/2007032. [30] J.-L. Lions and E. Magenes, Non-homogeneous blems and Applications. Vol. I, Translated from the French by P. Kenneth, Band 181, Springer-Verlag, New York-Heidelberg, 1972. [31] V. G. Maz'ja, Sobolev Spaces, Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3. [32] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [33] 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. [34] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. [35] A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017. [36] A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Applied Math., 65 (2012), 1242-1284. doi: 10.1002/cpa.21408. [37] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [38] X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations, Preprint, arXiv:1404.1197, 2014. [39] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. [40] L. N. Slobodeckij, Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations, Leningrad. Gos. Ped. Inst. Ućep. Zap., 197 (1958), 54-112. [41] J. Tan, Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-859. doi: 10.3934/dcds.2013.33.837. [42] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978; revised version 1995. [43] J. L. Vázquez, The dirichlet problem for the porous medium equation in bounded domains. Asymptotic behaviour, Monatsh. Math., 142 (2004), 81-111. doi: 10.1007/s00605-004-0237-4. [44] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. [45] J. L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type, J. Eur. Math. Soc., 16 (2014), 769-803. doi: 10.4171/JEMS/446. [46] J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, in Nonlinear Partial Differential Equations, Abel Symp., 7, Springer, Heidelberg, 2012, 271-298. doi: 10.1007/978-3-642-25361-4_15. [47] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Disc. Cont. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.

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##### References:
 [1] R. A. Adams and J. F. Fournier, Sobolev spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. [2] D. G. Aronson and L. A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains, J. Diff. Equations, 39 (1981), 378-412. doi: 10.1016/0022-0396(81)90065-6. [3] N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Tech. Report of Univ. of Kansas, 14 (1955), 77-94. [4] I. Athanasopoulos and L. A. Caffarelli, Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315. doi: 10.1016/j.aim.2009.11.010. [5] P. Bénilan and M. G. Crandall, The continuous dependence on $\varphi$ of solutions of $u_t - \Delta \varphi (u) = 0$, Indiana Univ. Math. J., 30 (1981), 161-177. doi: 10.1512/iumj.1981.30.30014. [6] R. M. Blumenthal and R. K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math., 9 (1959), 399-408. doi: 10.2140/pjm.1959.9.399. [7] M. Bonforte, G. Grillo and J. L. Vázquez, Fast diffusion flow on manifolds of nonpositive curvature, J. Evol. Eq., 8 (2008), 99-128. doi: 10.1007/s00028-007-0345-4. [8] M. Bonforte, G. Grillo and J. L. Vázquez, Behaviour near extinction for the fast diffusion equation on bounded domains, J. Math. Pures Appl., 97 (2012), 1-38. doi: 10.1016/j.matpur.2011.03.002. [9] M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Advances in Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018. [10] M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Arch. Ration. Mech. Anal., appeared online. doi: 10.1007/s00205-015-0861-2. [11] M. Bonforte and J. L. Vázquez, Nonlinear Degenerate Diffusion Equations on bounded domains with Restricted Fractional Laplacian, in preparation, 2014. [12] H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), Acad. Press, 1971, 101-156. [13] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland, 1973. [14] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. [15] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eq., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [16] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilinear equations, Comm. Partial Diff. Eq., 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. [17] Z. Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal., 226 (2005), 90-113. doi: 10.1016/j.jfa.2005.05.004. [18] A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034. [19] B. Dahlberg and C. Kenig, Nonnegative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Amer. Math. Soc., 1 (1988), 401-412. doi: 10.1090/S0894-0347-1988-0928264-9. [20] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990. [21] M. Crandall and M. Pierre, Regularizing Effects for $u_t=A\varphi(u)$ in $\mathbbL^1$, J. Funct. Anal., 45 (1982), 194-212. doi: 10.1016/0022-1236(82)90018-0. [22] E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623721. [23] E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal., 59 (1984), 335-395. doi: 10.1016/0022-1236(84)90076-4. [24] E. Gagliardo, Proprietá di alcune classi di funzioni in piú variabili, Ric. Mat., 7 (1958), 102-137. [25] G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators, Adv. in Math., 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018. [26] T. Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist., 22 (2002), 419-441. [27] T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. [28] N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. [29] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374. doi: 10.2307/2007032. [30] J.-L. Lions and E. Magenes, Non-homogeneous blems and Applications. Vol. I, Translated from the French by P. Kenneth, Band 181, Springer-Verlag, New York-Heidelberg, 1972. [31] V. G. Maz'ja, Sobolev Spaces, Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3. [32] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [33] 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. [34] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. [35] A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017. [36] A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Applied Math., 65 (2012), 1242-1284. doi: 10.1002/cpa.21408. [37] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [38] X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations, Preprint, arXiv:1404.1197, 2014. [39] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. [40] L. N. Slobodeckij, Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations, Leningrad. Gos. Ped. Inst. Ućep. Zap., 197 (1958), 54-112. [41] J. Tan, Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-859. doi: 10.3934/dcds.2013.33.837. [42] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978; revised version 1995. [43] J. L. Vázquez, The dirichlet problem for the porous medium equation in bounded domains. Asymptotic behaviour, Monatsh. Math., 142 (2004), 81-111. doi: 10.1007/s00605-004-0237-4. [44] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. [45] J. L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type, J. Eur. Math. Soc., 16 (2014), 769-803. doi: 10.4171/JEMS/446. [46] J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, in Nonlinear Partial Differential Equations, Abel Symp., 7, Springer, Heidelberg, 2012, 271-298. doi: 10.1007/978-3-642-25361-4_15. [47] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Disc. Cont. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.
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