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 & Continuous Dynamical Systems - A, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725
References:
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R. A. Adams and J. F. Fournier, Sobolev spaces,, Second edition, (2003). Google Scholar

[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. doi: 10.1016/0022-0396(81)90065-6. Google Scholar

[3]

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I. Athanasopoulos and L. A. Caffarelli, Continuity of the temperature in boundary heat control problems,, Adv. Math., 224 (2010), 293. doi: 10.1016/j.aim.2009.11.010. Google Scholar

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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. doi: 10.1512/iumj.1981.30.30014. Google Scholar

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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. doi: 10.2140/pjm.1959.9.399. Google Scholar

[7]

M. Bonforte, G. Grillo and J. L. Vázquez, Fast diffusion flow on manifolds of nonpositive curvature,, J. Evol. Eq., 8 (2008), 99. doi: 10.1007/s00028-007-0345-4. Google Scholar

[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. doi: 10.1016/j.matpur.2011.03.002. Google Scholar

[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. doi: 10.1016/j.aim.2013.09.018. Google Scholar

[10]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains,, Arch. Ration. Mech. Anal., (). doi: 10.1007/s00205-015-0861-2. Google Scholar

[11]

M. Bonforte and J. L. Vázquez, Nonlinear Degenerate Diffusion Equations on bounded domains with Restricted Fractional Laplacian,, in preparation, (2014). Google Scholar

[12]

H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations,, in Contributions to Nonlinear Functional Analysis (Proc. Sympos., (1971), 101. Google Scholar

[13]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert,, North-Holland, (1973). Google Scholar

[14]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[15]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Diff. Eq., 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[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. doi: 10.1080/03605302.2011.562954. Google Scholar

[17]

Z. Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains,, J. Funct. Anal., 226 (2005), 90. doi: 10.1016/j.jfa.2005.05.004. Google Scholar

[18]

A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives,, J. Math. Anal. Appl., 295 (2004), 225. doi: 10.1016/j.jmaa.2004.03.034. Google Scholar

[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. doi: 10.1090/S0894-0347-1988-0928264-9. Google Scholar

[20]

E. B. Davies, Heat Kernels and Spectral Theory,, Cambridge Tracts in Mathematics, (1990). Google Scholar

[21]

M. Crandall and M. Pierre, Regularizing Effects for $u_t=A\varphi(u)$ in $\mathbbL^1$,, J. Funct. Anal., 45 (1982), 194. doi: 10.1016/0022-1236(82)90018-0. Google Scholar

[22]

E. B. Davies, Spectral Theory and Differential Operators,, Cambridge Studies in Advanced Mathematics, (1995). doi: 10.1017/CBO9780511623721. Google Scholar

[23]

E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians,, J. Funct. Anal., 59 (1984), 335. doi: 10.1016/0022-1236(84)90076-4. Google Scholar

[24]

E. Gagliardo, Proprietá di alcune classi di funzioni in piú variabili,, Ric. Mat., 7 (1958), 102. Google Scholar

[25]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators,, Adv. in Math., 268 (2015), 478. doi: 10.1016/j.aim.2014.09.018. Google Scholar

[26]

T. Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes,, Probab. Math. Statist., 22 (2002), 419. Google Scholar

[27]

T. Kulczycki, Properties of Green function of symmetric stable processes,, Probab. Math. Statist., 17 (1997), 339. Google Scholar

[28]

N. S. Landkof, Foundations of Modern Potential Theory,, Die Grundlehren der mathematischen Wissenschaften, (1972). Google Scholar

[29]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math. (2), 118 (1983), 349. doi: 10.2307/2007032. Google Scholar

[30]

J.-L. Lions and E. Magenes, Non-homogeneous blems and Applications. Vol. I,, Translated from the French by P. Kenneth, (1972). Google Scholar

[31]

V. G. Maz'ja, Sobolev Spaces,, Translated from the Russian by T. O. Shaposhnikova, (1985). doi: 10.1007/978-3-662-09922-3. Google Scholar

[32]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar

[33]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[34]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115. Google Scholar

[35]

A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A fractional porous medium equation,, Adv. Math., 226 (2011), 1378. doi: 10.1016/j.aim.2010.07.017. Google Scholar

[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. doi: 10.1002/cpa.21408. Google Scholar

[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. doi: 10.1016/j.matpur.2013.06.003. Google Scholar

[38]

X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations,, Preprint, (2014). Google Scholar

[39]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators,, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831. doi: 10.1017/S0308210512001783. Google Scholar

[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. Google Scholar

[41]

J. Tan, Positive solutions for non local elliptic problems,, Discrete Contin. Dyn. Syst., 33 (2013), 837. doi: 10.3934/dcds.2013.33.837. Google Scholar

[42]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Publishing Company, (1978). Google Scholar

[43]

J. L. Vázquez, The dirichlet problem for the porous medium equation in bounded domains. Asymptotic behaviour,, Monatsh. Math., 142 (2004), 81. doi: 10.1007/s00605-004-0237-4. Google Scholar

[44]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs, (2007). Google Scholar

[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. doi: 10.4171/JEMS/446. Google Scholar

[46]

J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators,, in Nonlinear Partial Differential Equations, (2012), 271. doi: 10.1007/978-3-642-25361-4_15. Google Scholar

[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. doi: 10.3934/dcdss.2014.7.857. Google Scholar

show all references

References:
[1]

R. A. Adams and J. F. Fournier, Sobolev spaces,, Second edition, (2003). Google Scholar

[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. doi: 10.1016/0022-0396(81)90065-6. Google Scholar

[3]

N. Aronszajn, Boundary values of functions with finite Dirichlet integral,, Tech. Report of Univ. of Kansas, 14 (1955), 77. Google Scholar

[4]

I. Athanasopoulos and L. A. Caffarelli, Continuity of the temperature in boundary heat control problems,, Adv. Math., 224 (2010), 293. doi: 10.1016/j.aim.2009.11.010. Google Scholar

[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. doi: 10.1512/iumj.1981.30.30014. Google Scholar

[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. doi: 10.2140/pjm.1959.9.399. Google Scholar

[7]

M. Bonforte, G. Grillo and J. L. Vázquez, Fast diffusion flow on manifolds of nonpositive curvature,, J. Evol. Eq., 8 (2008), 99. doi: 10.1007/s00028-007-0345-4. Google Scholar

[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. doi: 10.1016/j.matpur.2011.03.002. Google Scholar

[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. doi: 10.1016/j.aim.2013.09.018. Google Scholar

[10]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains,, Arch. Ration. Mech. Anal., (). doi: 10.1007/s00205-015-0861-2. Google Scholar

[11]

M. Bonforte and J. L. Vázquez, Nonlinear Degenerate Diffusion Equations on bounded domains with Restricted Fractional Laplacian,, in preparation, (2014). Google Scholar

[12]

H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations,, in Contributions to Nonlinear Functional Analysis (Proc. Sympos., (1971), 101. Google Scholar

[13]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert,, North-Holland, (1973). Google Scholar

[14]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[15]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Diff. Eq., 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[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. doi: 10.1080/03605302.2011.562954. Google Scholar

[17]

Z. Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains,, J. Funct. Anal., 226 (2005), 90. doi: 10.1016/j.jfa.2005.05.004. Google Scholar

[18]

A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives,, J. Math. Anal. Appl., 295 (2004), 225. doi: 10.1016/j.jmaa.2004.03.034. Google Scholar

[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. doi: 10.1090/S0894-0347-1988-0928264-9. Google Scholar

[20]

E. B. Davies, Heat Kernels and Spectral Theory,, Cambridge Tracts in Mathematics, (1990). Google Scholar

[21]

M. Crandall and M. Pierre, Regularizing Effects for $u_t=A\varphi(u)$ in $\mathbbL^1$,, J. Funct. Anal., 45 (1982), 194. doi: 10.1016/0022-1236(82)90018-0. Google Scholar

[22]

E. B. Davies, Spectral Theory and Differential Operators,, Cambridge Studies in Advanced Mathematics, (1995). doi: 10.1017/CBO9780511623721. Google Scholar

[23]

E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians,, J. Funct. Anal., 59 (1984), 335. doi: 10.1016/0022-1236(84)90076-4. Google Scholar

[24]

E. Gagliardo, Proprietá di alcune classi di funzioni in piú variabili,, Ric. Mat., 7 (1958), 102. Google Scholar

[25]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators,, Adv. in Math., 268 (2015), 478. doi: 10.1016/j.aim.2014.09.018. Google Scholar

[26]

T. Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes,, Probab. Math. Statist., 22 (2002), 419. Google Scholar

[27]

T. Kulczycki, Properties of Green function of symmetric stable processes,, Probab. Math. Statist., 17 (1997), 339. Google Scholar

[28]

N. S. Landkof, Foundations of Modern Potential Theory,, Die Grundlehren der mathematischen Wissenschaften, (1972). Google Scholar

[29]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math. (2), 118 (1983), 349. doi: 10.2307/2007032. Google Scholar

[30]

J.-L. Lions and E. Magenes, Non-homogeneous blems and Applications. Vol. I,, Translated from the French by P. Kenneth, (1972). Google Scholar

[31]

V. G. Maz'ja, Sobolev Spaces,, Translated from the Russian by T. O. Shaposhnikova, (1985). doi: 10.1007/978-3-662-09922-3. Google Scholar

[32]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar

[33]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[34]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115. Google Scholar

[35]

A. de Pablo, F. Quirós, A. Rodriguez and J. L. Vázquez, A fractional porous medium equation,, Adv. Math., 226 (2011), 1378. doi: 10.1016/j.aim.2010.07.017. Google Scholar

[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. doi: 10.1002/cpa.21408. Google Scholar

[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. doi: 10.1016/j.matpur.2013.06.003. Google Scholar

[38]

X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations,, Preprint, (2014). Google Scholar

[39]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators,, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831. doi: 10.1017/S0308210512001783. Google Scholar

[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. Google Scholar

[41]

J. Tan, Positive solutions for non local elliptic problems,, Discrete Contin. Dyn. Syst., 33 (2013), 837. doi: 10.3934/dcds.2013.33.837. Google Scholar

[42]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Publishing Company, (1978). Google Scholar

[43]

J. L. Vázquez, The dirichlet problem for the porous medium equation in bounded domains. Asymptotic behaviour,, Monatsh. Math., 142 (2004), 81. doi: 10.1007/s00605-004-0237-4. Google Scholar

[44]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs, (2007). Google Scholar

[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. doi: 10.4171/JEMS/446. Google Scholar

[46]

J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators,, in Nonlinear Partial Differential Equations, (2012), 271. doi: 10.1007/978-3-642-25361-4_15. Google Scholar

[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. doi: 10.3934/dcdss.2014.7.857. Google Scholar

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