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

References:
[1]

Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

J. Diff. Equations, 39 (1981), 378-412. doi: 10.1016/0022-0396(81)90065-6.  Google Scholar

[3]

Tech. Report of Univ. of Kansas, 14 (1955), 77-94. Google Scholar

[4]

Adv. Math., 224 (2010), 293-315. doi: 10.1016/j.aim.2009.11.010.  Google Scholar

[5]

Indiana Univ. Math. J., 30 (1981), 161-177. doi: 10.1512/iumj.1981.30.30014.  Google Scholar

[6]

Pacific J. Math., 9 (1959), 399-408. doi: 10.2140/pjm.1959.9.399.  Google Scholar

[7]

J. Evol. Eq., 8 (2008), 99-128. doi: 10.1007/s00028-007-0345-4.  Google Scholar

[8]

J. Math. Pures Appl., 97 (2012), 1-38. doi: 10.1016/j.matpur.2011.03.002.  Google Scholar

[9]

Advances in Math., 250 (2014), 242-284. 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]

in preparation, 2014. Google Scholar

[12]

in Contributions to Nonlinear Functional Analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), Acad. Press, 1971, 101-156.  Google Scholar

[13]

North-Holland, 1973.  Google Scholar

[14]

Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[15]

Comm. Partial Diff. Eq., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar

[16]

Comm. Partial Diff. Eq., 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954.  Google Scholar

[17]

J. Funct. Anal., 226 (2005), 90-113. doi: 10.1016/j.jfa.2005.05.004.  Google Scholar

[18]

J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034.  Google Scholar

[19]

J. Amer. Math. Soc., 1 (1988), 401-412. doi: 10.1090/S0894-0347-1988-0928264-9.  Google Scholar

[20]

Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990.  Google Scholar

[21]

J. Funct. Anal., 45 (1982), 194-212. doi: 10.1016/0022-1236(82)90018-0.  Google Scholar

[22]

Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623721.  Google Scholar

[23]

J. Funct. Anal., 59 (1984), 335-395. doi: 10.1016/0022-1236(84)90076-4.  Google Scholar

[24]

Ric. Mat., 7 (1958), 102-137.  Google Scholar

[25]

Adv. in Math., 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018.  Google Scholar

[26]

Probab. Math. Statist., 22 (2002), 419-441.  Google Scholar

[27]

Probab. Math. Statist., 17 (1997), 339-364.  Google Scholar

[28]

Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[29]

Ann. of Math. (2), 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[30]

Translated from the French by P. Kenneth, Band 181, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[31]

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.  Google Scholar

[32]

Cambridge University Press, Cambridge, 2000.  Google Scholar

[33]

Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[34]

Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162.  Google Scholar

[35]

Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017.  Google Scholar

[36]

Comm. Pure Applied Math., 65 (2012), 1242-1284. doi: 10.1002/cpa.21408.  Google Scholar

[37]

J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[38]

Preprint, arXiv:1404.1197, 2014. Google Scholar

[39]

Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783.  Google Scholar

[40]

Leningrad. Gos. Ped. Inst. Ućep. Zap., 197 (1958), 54-112.  Google Scholar

[41]

Discrete Contin. Dyn. Syst., 33 (2013), 837-859. doi: 10.3934/dcds.2013.33.837.  Google Scholar

[42]

North-Holland Publishing Company, Amsterdam, New York, Oxford, 1978; revised version 1995.  Google Scholar

[43]

Monatsh. Math., 142 (2004), 81-111. doi: 10.1007/s00605-004-0237-4.  Google Scholar

[44]

Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007.  Google Scholar

[45]

J. Eur. Math. Soc., 16 (2014), 769-803. doi: 10.4171/JEMS/446.  Google Scholar

[46]

in Nonlinear Partial Differential Equations, Abel Symp., 7, Springer, Heidelberg, 2012, 271-298. doi: 10.1007/978-3-642-25361-4_15.  Google Scholar

[47]

Disc. Cont. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.  Google Scholar

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