August  2014, 7(4): 857-885. doi: 10.3934/dcdss.2014.7.857

Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators

1. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid

Received  November 2013 Published  February 2014

We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual porous medium flows, the fractional version has infinite speed of propagation for all exponents $0 < s < 1$ and $m > 0$; on the other hand, it also generates an $L^1$-contraction semigroup which depends continuously on the exponent of fractional differentiation and the exponent of the nonlinearity.
    After establishing the general existence and uniqueness theory, the main properties are described: positivity, regularity, continuous dependence, a priori estimates, Schwarz symmetrization, among others. Self-similar solutions are constructed (fractional Barenblatt solutions) and they explain the asymptotic behaviour of a large class of solutions. In the fast diffusion range we study extinction in finite time and we find suitable special solutions. We discuss KPP type propagation. We also examine some related equations that extend the model and briefly comment on current work.
Citation: Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857
References:
[1]

SIAM J. Math. Anal., 44 (2012), 603-632. doi: 10.1137/110834342.  Google Scholar

[2]

N. Alibaud, S. Cifani and E. Jakobsen, Optimal continuous dependence estimates for fractional degenerate parabolic equations,, , ().   Google Scholar

[3]

Annales IHP, Analyse non linéaire, 28 (2011), 217-246. doi: 10.1016/j.anihpc.2010.11.006.  Google Scholar

[4]

Comm. Pure Appl. Math., 61 (2008), 1495-1539. doi: 10.1002/cpa.20223.  Google Scholar

[5]

AMS Mathematical Surveys and Monographs, 165, 2010. Google Scholar

[6]

Second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[7]

in Nonlinear Diffusion Problems (Montecatini Terme, 1985), Lecture Notes in Math., 1224, Springer, Berlin, 1986, 1-46. doi: 10.1007/BFb0072687.  Google Scholar

[8]

Trans. Amer. Math. Soc., 280 (1983), 351-366. doi: 10.1090/S0002-9947-1983-0712265-1.  Google Scholar

[9]

Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[10]

Zap. Nauchn. Se. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts, 35 [34], 49-66, 226; translation in J. Math. Sci. (N. Y.), 132 (2006), 274-284. doi: 10.1007/s10958-005-0496-1.  Google Scholar

[11]

Advances in Mathematics, 224 (2010), 293-315. doi: 10.1016/j.aim.2009.11.010.  Google Scholar

[12]

J. Diff. Geom., 11 (1976), 573-598.  Google Scholar

[13]

Monographs and Studies in Mathematics, 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980.  Google Scholar

[14]

(in Russian) Prikl. Mat. Mekh., 16 (1952), 67-78.  Google Scholar

[15]

Cambridge Texts in Applied Mathematics, 14, Cambridge University Press, Cambridge, 1996.  Google Scholar

[16]

Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.  Google Scholar

[17]

Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009.  Google Scholar

[18]

M3AS, 22 (2012), 1140005, 39 pp. Google Scholar

[19]

Comptes Rendus Mathematique, 349(2011), 641-645. doi: 10.1016/j.crma.2011.06.003.  Google Scholar

[20]

preprint arXiv:1302.7219, (2013). Google Scholar

[21]

Comm. Math. Phys., 294 (2010), 145-168. doi: 10.1007/s00220-009-0855-8.  Google Scholar

[22]

Trans. Amer. Math. Soc., 95 (1960), 263-273. doi: 10.1090/S0002-9947-1960-0119247-6.  Google Scholar

[23]

Physical Review E, 62 (2000). doi: 10.1103/PhysRevE.62.2213.  Google Scholar

[24]

J. Funct. Anal., 240 (2006), 399-428. doi: 10.1016/j.jfa.2006.07.009.  Google Scholar

[25]

Advances in Math., 223 (2010), 529-578. doi: 10.1016/j.aim.2009.08.021.  Google Scholar

[26]

Proceedings Natl. Acad. Sci. USA, 107 (2010), 16459-16464. doi: 10.1073/pnas.1003972107.  Google Scholar

[27]

Advances in Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018.  Google Scholar

[28]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains,, preprint , ().   Google Scholar

[29]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains. Part II,, in preparation., ().   Google Scholar

[30]

J. Anal. Math., 87 (2002), 77-101  Google Scholar

[31]

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

[32]

C. R. Math. Acad. Sci. Paris, 347 (2009), 1361-1366. doi: 10.1016/j.crma.2009.10.012.  Google Scholar

[33]

Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5.  Google Scholar

[34]

Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.  Google Scholar

[35]

J. Amer. Math. Soc., 24 (2011), 849-869. doi: 10.1090/S0894-0347-2011-00698-X.  Google Scholar

[36]

Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.  Google Scholar

[37]

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

[38]

J. Eur. Math. Soc. (JEMS), 15 (2013), 1701-1746. doi: 10.4171/JEMS/401.  Google Scholar

[39]

Arch. Rational Mech. Anal., 202 (2011), 537-565. doi: 10.1007/s00205-011-0420-4.  Google Scholar

[40]

Discrete Cont. Dyn. Systems-A, 29 (2011), 1393-1404.  Google Scholar

[41]

Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.  Google Scholar

[42]

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

[43]

J. A. Carrillo, Y. Huang and J. L. Vazquez, in, preparation., ().   Google Scholar

[44]

J. Eur. Math. Soc. (JEMS), 12 (2010), 1307-1329. doi: 10.4171/JEMS/231.  Google Scholar

[45]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 413-441. doi: 10.1016/j.anihpc.2011.02.006.  Google Scholar

[46]

BIT, 51 (2011), 809-844. doi: 10.1007/s10543-011-0327-3.  Google Scholar

[47]

Eur. J. Appl. Math., 7 (1996), 97-111. doi: 10.1017/S0956792500002242.  Google Scholar

[48]

Chapman & Hall/CRC, 2004.  Google Scholar

[49]

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

[50]

J. Differential Equations, 93 (1991), 19-61. doi: 10.1016/0022-0396(91)90021-Z.  Google Scholar

[51]

Advances in Mathematics, 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017.  Google Scholar

[52]

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

[53]

A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, Classical solutions for a logarithmic fractional diffusion equation,, to appear in Journal de Math. Pures Appliquées, ().   Google Scholar

[54]

preprint, (2011). Google Scholar

[55]

Travaux et Recherches Mathématiques, No. 21, Dunod, Paris, 1972.  Google Scholar

[56]

Phys. Rev. B, 50 (1994), 1126-1135. Google Scholar

[57]

Ann. Eugenics, 7 (1937), 355-369. Google Scholar

[58]

Trans. Amer. Math. Soc., 101 (1961), 75-90. doi: 10.1090/S0002-9947-1961-0137148-5.  Google Scholar

[59]

Phil. Mag., 26 (1972), 65-72. Google Scholar

[60]

preprint, (2013). Google Scholar

[61]

Comm. Pure Applied Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253.  Google Scholar

[62]

Ann. Appl. Probab., 19 (2009), 2270-2300. doi: 10.1214/09-AAP610.  Google Scholar

[63]

M. Jara, Hydrodynamic limit Of particle systems with long jumps,, , ().   Google Scholar

[64]

Probab. Theory Relat. Fields, 145 (2009), 565-590. doi: 10.1007/s00440-008-0178-2.  Google Scholar

[65]

Calc. Var., 34 (2009), 1-21. doi: 10.1007/s00526-008-0173-6.  Google Scholar

[66]

R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2529-2546. doi: 10.1098/rspa.2003.1134.  Google Scholar

[67]

Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.  Google Scholar

[68]

Bjul. Moskowskogo Gos. Univ., 17 (1937), 1-26. Google Scholar

[69]

Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer, New York, 1972.  Google Scholar

[70]

Physical Review E, 67 (2003), 031104. doi: 10.1103/PhysRevE.67.031104.  Google Scholar

[71]

Discrete Cont. Dyn. Systems, 6 (2000), 121-142. doi: 10.3934/dcds.2000.6.121.  Google Scholar

[72]

Journal Funct. Anal., 195 (2002), 230-238. doi: 10.1006/jfan.2002.3955.  Google Scholar

[73]

Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.  Google Scholar

[74]

Physics Reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[75]

R. H. Nochetto, E. Otarola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis,, , ().   Google Scholar

[76]

Calc. Var. PDEs, online, arXiv:1205.632229, (2013). doi: 10.1007/s00526-013-0613-9.  Google Scholar

[77]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators,, to appear in Proc. Roy. Soc. Edinburgh Sect. A. Available from: , (): 12.   Google Scholar

[78]

Rendiconti di Matematica e delle sue Applicazioni, 18 (1959), 95-139.  Google Scholar

[79]

Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706.  Google Scholar

[80]

Comm. Pure Appl. Math., 60 (2007), 6-112. doi: 10.1002/cpa.20153.  Google Scholar

[81]

submitted, arXiv:1303.6823, (2013). Google Scholar

[82]

Comptes Rendus Acad. Sci. Paris, 352 (2014), 123-128, arXiv:1311.7007. Google Scholar

[83]

Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[84]

Ann. Scuola Norm. Sup. (4), 3 (1976), 697-718.  Google Scholar

[85]

Ann. Mat. Pura Appl. (4), 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[86]

to appear in Calcolo, arXiv:1301.4349, (2013). doi: 10.1007/s10092-013-0103-7.  Google Scholar

[87]

F. del Teso and J. L. Vázquez, Finite difference method for a general fractional porous medium equation,, , ().   Google Scholar

[88]

Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.  Google Scholar

[89]

C. R. Acad. Sc. Paris, 295 (1982), 71-74.  Google Scholar

[90]

Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar

[91]

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

[92]

in Nonlinear partial differential equations: the Abel Symposium 2010 (ed. H. Kenneth), Holden, Helge & Karlsen, Springer, 2012, 271-298. Google Scholar

[93]

to appear in J. Europ. Math. Soc.; arXiv:1205.6332, (2013). Google Scholar

[94]

J. L. Vázquez, A. de Pablo, F. Quirós and A. Rodríguez, Classical solutions and higher regularity for nonlinear fractional diffusion equations;, , ().   Google Scholar

[95]

J. L. Vázquez and B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type,, to appear in J. Math. Pures Appl.; , ().   Google Scholar

[96]

J. L. Vázquez and B. Volzone, Optimal estimates for Fractional Fast diffusion equations,, submitted, ().   Google Scholar

[97]

in Order and Chaos, 10th volume, (ed. T. Bountis), Patras University Press, 2008. Google Scholar

[98]

in 1962 Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press, Stanford, Calif., 1962, 424-428.  Google Scholar

[99]

Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281. doi: 10.1016/S1007-5704(03)00049-2.  Google Scholar

[100]

in Lévy Processes - Theory and Applications, (eds. T. Mikosch, O. Barndorff-Nielsen and S. Resnick), Birkhäuser, Boston, 2001, 241-266.  Google Scholar

[101]

in Collection of Papers Dedicated to 70th Anniversary of A. F. Ioffe, Izd. Akad. Nauk SSSR, Moscow, 1950, 61-72. Google Scholar

show all references

References:
[1]

SIAM J. Math. Anal., 44 (2012), 603-632. doi: 10.1137/110834342.  Google Scholar

[2]

N. Alibaud, S. Cifani and E. Jakobsen, Optimal continuous dependence estimates for fractional degenerate parabolic equations,, , ().   Google Scholar

[3]

Annales IHP, Analyse non linéaire, 28 (2011), 217-246. doi: 10.1016/j.anihpc.2010.11.006.  Google Scholar

[4]

Comm. Pure Appl. Math., 61 (2008), 1495-1539. doi: 10.1002/cpa.20223.  Google Scholar

[5]

AMS Mathematical Surveys and Monographs, 165, 2010. Google Scholar

[6]

Second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[7]

in Nonlinear Diffusion Problems (Montecatini Terme, 1985), Lecture Notes in Math., 1224, Springer, Berlin, 1986, 1-46. doi: 10.1007/BFb0072687.  Google Scholar

[8]

Trans. Amer. Math. Soc., 280 (1983), 351-366. doi: 10.1090/S0002-9947-1983-0712265-1.  Google Scholar

[9]

Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[10]

Zap. Nauchn. Se. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts, 35 [34], 49-66, 226; translation in J. Math. Sci. (N. Y.), 132 (2006), 274-284. doi: 10.1007/s10958-005-0496-1.  Google Scholar

[11]

Advances in Mathematics, 224 (2010), 293-315. doi: 10.1016/j.aim.2009.11.010.  Google Scholar

[12]

J. Diff. Geom., 11 (1976), 573-598.  Google Scholar

[13]

Monographs and Studies in Mathematics, 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980.  Google Scholar

[14]

(in Russian) Prikl. Mat. Mekh., 16 (1952), 67-78.  Google Scholar

[15]

Cambridge Texts in Applied Mathematics, 14, Cambridge University Press, Cambridge, 1996.  Google Scholar

[16]

Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.  Google Scholar

[17]

Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009.  Google Scholar

[18]

M3AS, 22 (2012), 1140005, 39 pp. Google Scholar

[19]

Comptes Rendus Mathematique, 349(2011), 641-645. doi: 10.1016/j.crma.2011.06.003.  Google Scholar

[20]

preprint arXiv:1302.7219, (2013). Google Scholar

[21]

Comm. Math. Phys., 294 (2010), 145-168. doi: 10.1007/s00220-009-0855-8.  Google Scholar

[22]

Trans. Amer. Math. Soc., 95 (1960), 263-273. doi: 10.1090/S0002-9947-1960-0119247-6.  Google Scholar

[23]

Physical Review E, 62 (2000). doi: 10.1103/PhysRevE.62.2213.  Google Scholar

[24]

J. Funct. Anal., 240 (2006), 399-428. doi: 10.1016/j.jfa.2006.07.009.  Google Scholar

[25]

Advances in Math., 223 (2010), 529-578. doi: 10.1016/j.aim.2009.08.021.  Google Scholar

[26]

Proceedings Natl. Acad. Sci. USA, 107 (2010), 16459-16464. doi: 10.1073/pnas.1003972107.  Google Scholar

[27]

Advances in Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018.  Google Scholar

[28]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains,, preprint , ().   Google Scholar

[29]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains. Part II,, in preparation., ().   Google Scholar

[30]

J. Anal. Math., 87 (2002), 77-101  Google Scholar

[31]

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

[32]

C. R. Math. Acad. Sci. Paris, 347 (2009), 1361-1366. doi: 10.1016/j.crma.2009.10.012.  Google Scholar

[33]

Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5.  Google Scholar

[34]

Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.  Google Scholar

[35]

J. Amer. Math. Soc., 24 (2011), 849-869. doi: 10.1090/S0894-0347-2011-00698-X.  Google Scholar

[36]

Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.  Google Scholar

[37]

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

[38]

J. Eur. Math. Soc. (JEMS), 15 (2013), 1701-1746. doi: 10.4171/JEMS/401.  Google Scholar

[39]

Arch. Rational Mech. Anal., 202 (2011), 537-565. doi: 10.1007/s00205-011-0420-4.  Google Scholar

[40]

Discrete Cont. Dyn. Systems-A, 29 (2011), 1393-1404.  Google Scholar

[41]

Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.  Google Scholar

[42]

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

[43]

J. A. Carrillo, Y. Huang and J. L. Vazquez, in, preparation., ().   Google Scholar

[44]

J. Eur. Math. Soc. (JEMS), 12 (2010), 1307-1329. doi: 10.4171/JEMS/231.  Google Scholar

[45]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 413-441. doi: 10.1016/j.anihpc.2011.02.006.  Google Scholar

[46]

BIT, 51 (2011), 809-844. doi: 10.1007/s10543-011-0327-3.  Google Scholar

[47]

Eur. J. Appl. Math., 7 (1996), 97-111. doi: 10.1017/S0956792500002242.  Google Scholar

[48]

Chapman & Hall/CRC, 2004.  Google Scholar

[49]

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

[50]

J. Differential Equations, 93 (1991), 19-61. doi: 10.1016/0022-0396(91)90021-Z.  Google Scholar

[51]

Advances in Mathematics, 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017.  Google Scholar

[52]

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

[53]

A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, Classical solutions for a logarithmic fractional diffusion equation,, to appear in Journal de Math. Pures Appliquées, ().   Google Scholar

[54]

preprint, (2011). Google Scholar

[55]

Travaux et Recherches Mathématiques, No. 21, Dunod, Paris, 1972.  Google Scholar

[56]

Phys. Rev. B, 50 (1994), 1126-1135. Google Scholar

[57]

Ann. Eugenics, 7 (1937), 355-369. Google Scholar

[58]

Trans. Amer. Math. Soc., 101 (1961), 75-90. doi: 10.1090/S0002-9947-1961-0137148-5.  Google Scholar

[59]

Phil. Mag., 26 (1972), 65-72. Google Scholar

[60]

preprint, (2013). Google Scholar

[61]

Comm. Pure Applied Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253.  Google Scholar

[62]

Ann. Appl. Probab., 19 (2009), 2270-2300. doi: 10.1214/09-AAP610.  Google Scholar

[63]

M. Jara, Hydrodynamic limit Of particle systems with long jumps,, , ().   Google Scholar

[64]

Probab. Theory Relat. Fields, 145 (2009), 565-590. doi: 10.1007/s00440-008-0178-2.  Google Scholar

[65]

Calc. Var., 34 (2009), 1-21. doi: 10.1007/s00526-008-0173-6.  Google Scholar

[66]

R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2529-2546. doi: 10.1098/rspa.2003.1134.  Google Scholar

[67]

Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.  Google Scholar

[68]

Bjul. Moskowskogo Gos. Univ., 17 (1937), 1-26. Google Scholar

[69]

Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer, New York, 1972.  Google Scholar

[70]

Physical Review E, 67 (2003), 031104. doi: 10.1103/PhysRevE.67.031104.  Google Scholar

[71]

Discrete Cont. Dyn. Systems, 6 (2000), 121-142. doi: 10.3934/dcds.2000.6.121.  Google Scholar

[72]

Journal Funct. Anal., 195 (2002), 230-238. doi: 10.1006/jfan.2002.3955.  Google Scholar

[73]

Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.  Google Scholar

[74]

Physics Reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[75]

R. H. Nochetto, E. Otarola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis,, , ().   Google Scholar

[76]

Calc. Var. PDEs, online, arXiv:1205.632229, (2013). doi: 10.1007/s00526-013-0613-9.  Google Scholar

[77]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators,, to appear in Proc. Roy. Soc. Edinburgh Sect. A. Available from: , (): 12.   Google Scholar

[78]

Rendiconti di Matematica e delle sue Applicazioni, 18 (1959), 95-139.  Google Scholar

[79]

Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706.  Google Scholar

[80]

Comm. Pure Appl. Math., 60 (2007), 6-112. doi: 10.1002/cpa.20153.  Google Scholar

[81]

submitted, arXiv:1303.6823, (2013). Google Scholar

[82]

Comptes Rendus Acad. Sci. Paris, 352 (2014), 123-128, arXiv:1311.7007. Google Scholar

[83]

Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[84]

Ann. Scuola Norm. Sup. (4), 3 (1976), 697-718.  Google Scholar

[85]

Ann. Mat. Pura Appl. (4), 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[86]

to appear in Calcolo, arXiv:1301.4349, (2013). doi: 10.1007/s10092-013-0103-7.  Google Scholar

[87]

F. del Teso and J. L. Vázquez, Finite difference method for a general fractional porous medium equation,, , ().   Google Scholar

[88]

Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.  Google Scholar

[89]

C. R. Acad. Sc. Paris, 295 (1982), 71-74.  Google Scholar

[90]

Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar

[91]

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

[92]

in Nonlinear partial differential equations: the Abel Symposium 2010 (ed. H. Kenneth), Holden, Helge & Karlsen, Springer, 2012, 271-298. Google Scholar

[93]

to appear in J. Europ. Math. Soc.; arXiv:1205.6332, (2013). Google Scholar

[94]

J. L. Vázquez, A. de Pablo, F. Quirós and A. Rodríguez, Classical solutions and higher regularity for nonlinear fractional diffusion equations;, , ().   Google Scholar

[95]

J. L. Vázquez and B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type,, to appear in J. Math. Pures Appl.; , ().   Google Scholar

[96]

J. L. Vázquez and B. Volzone, Optimal estimates for Fractional Fast diffusion equations,, submitted, ().   Google Scholar

[97]

in Order and Chaos, 10th volume, (ed. T. Bountis), Patras University Press, 2008. Google Scholar

[98]

in 1962 Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press, Stanford, Calif., 1962, 424-428.  Google Scholar

[99]

Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281. doi: 10.1016/S1007-5704(03)00049-2.  Google Scholar

[100]

in Lévy Processes - Theory and Applications, (eds. T. Mikosch, O. Barndorff-Nielsen and S. Resnick), Birkhäuser, Boston, 2001, 241-266.  Google Scholar

[101]

in Collection of Papers Dedicated to 70th Anniversary of A. F. Ioffe, Izd. Akad. Nauk SSSR, Moscow, 1950, 61-72. Google Scholar

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