December  2015, 35(12): 5927-5962. doi: 10.3934/dcds.2015.35.5927

On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

1. 

Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

2. 

Dipartimento di Matematica , Università di Milano, via Cesare Saldini 50, 20133 Milano, Italy

Received  March 2014 Published  May 2015

We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation.
Citation: 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
References:
[1]

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.

[2]

P. Bénilan and M. G. Crandall, The continuous dependence on $\phi$ of solutions of $u_t-\Delta\phi(u)=0 $, Indiana Univ. Math. J., 30 (1981), 161-177. doi: 10.1512/iumj.1981.30.30014.

[3]

P. Bénilan and R. Gariepy, Strong solutions in $L^1$ of degenerate parabolic equations, J. Differential Equations, 119 (1995), 473-502. doi: 10.1006/jdeq.1995.1099.

[4]

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Rat. Mech. Anal., 191 (2009), 347-385. doi: 10.1007/s00205-008-0155-z.

[5]

M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, preprint, arXiv:1404.6195, 2014.

[6]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, preprint, arXiv:1311.6997, 2013. doi: 10.1007/s00205-015-0861-2.

[7]

M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018.

[8]

C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.

[9]

H. Brezis and S. Kamin, Sublinear Elliptic Equations in $\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.

[10]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.

[11]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[12]

W. Choi, S. Kim and K.-A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598. doi: 10.1016/j.jfa.2014.02.029.

[13]

A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017.

[14]

A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284. doi: 10.1002/cpa.21408.

[15]

E. Di Benedetto, Continuity of Weak Solutions to a General Porous Medium Equation, Indiana Univ. Math. J., 32 (1983), 83-118. doi: 10.1512/iumj.1983.32.32008.

[16]

G. Di Blasio and B. Volzone, Comparison and regularity results for the fractional Laplacian via symmetrization methods, J. Differential Equations, 253 (2012), 2593-2615. doi: 10.1016/j.jde.2012.07.004.

[17]

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.

[18]

D. Eidus, The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium, J. Differential Equations, 84 (1990), 309-318. doi: 10.1016/0022-0396(90)90081-Y.

[19]

D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830. doi: 10.1090/S0002-9939-1994-1169025-2.

[20]

A. Friedman and S. Kamin, The asymptotic behavior of gas in an $n$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563. doi: 10.2307/1999846.

[21]

G. Grillo, M. Muratori and M. M. Porzio, Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities, Discrete Contin. Dyn. Syst., 33 (2013), 3599-3640. doi: 10.3934/dcds.2013.33.3599.

[22]

G. Grillo, M. Muratori and F. Punzo, Conditions at infinity for the inhomogeneous filtration equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 413-428. doi: 10.1016/j.anihpc.2013.04.002.

[23]

G. Grillo, M. Muratori and F. Punzo, Weighted fractional porous media equations: Existence and uniqueness of weak solutions with measure data, preprint, arXiv:1312.6076, 2013.

[24]

R. G. Iagar and A. Sánchez, Asymptotic behavior for the heat equation in nonhomogeneous media with critical density, Nonlinear Anal., 89 (2013), 24-35. doi: 10.1016/j.na.2013.05.002.

[25]

R. G. Iagar, A. Sánchez, Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density, Nonlinear Anal., 102 (2014), 226-241. doi: 10.1016/j.na.2014.02.016.

[26]

S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 9 (1998), 279-298.

[27]

S. Kamin, G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, Discrete Contin. Dyn. Syst., 26 (2010), 521-549. doi: 10.3934/dcds.2010.26.521.

[28]

S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127. doi: 10.1002/cpa.3160350106.

[29]

S. Nieto and G. Reyes, Asymptotic behavior of the solutions of the inhomogeneous porous medium equation with critical vanishing density, Commun. Pure Appl. Anal., 12 (2013), 1123-1139. doi: 10.3934/cpaa.2013.12.1123.

[30]

M. Pierre, Uniqueness of the solutions of $u_t - \Delta \varphi (u)=0$ with initial datum a measure, Nonlinear Anal., 6 (1982), 175-187. doi: 10.1016/0362-546X(82)90086-4.

[31]

F. Punzo, On the Cauchy problem for nonlinear parabolic equations with variable density, J. Evol. Equ., 9 (2009), 429-447. doi: 10.1007/s00028-009-0018-6.

[32]

F. Punzo and G. Terrone, On the Cauchy problem for a general fractional porous medium equation with variable density, Nonlinear Anal., 98 (2014), 27-47. doi: 10.1016/j.na.2013.12.007.

[33]

F. Punzo and G. Terrone, Well-posedness for the Cauchy problem for a fractional porous medium equation with variable density in one space dimension, Differential Integral Equations, 27 (2014), 461-482.

[34]

F. Punzo and G. Terrone, On a fractional sublinear elliptic equation with a variable coefficient, Appl. Anal., 94 (2015), 800-818. doi: 10.1080/00036811.2014.902053.

[35]

F. Punzo and E. Valdinoci, Uniqueness in weighted Lebesgue spaces for a class of fractional elliptic and parabolic equations, J. Differential Equations, 258 (2015), 555-587. doi: 10.1016/j.jde.2014.09.023.

[36]

V. D. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods, Contemporary Mathematics and Its Applications, 6, Hindawi Publishing Corporation, New York, 2008. doi: 10.1155/9789774540394.

[37]

G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508. doi: 10.3934/cpaa.2009.8.493.

[38]

G. Reyes and J. L. Vázquez, The Cauchy problem for the inhomogeneous porous medium equation, Netw. Heterog. Media, 1 (2006), 337-351. doi: 10.3934/nhm.2006.1.337.

[39]

G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal., 7 (2008), 1275-1294. doi: 10.3934/cpaa.2008.7.1275.

[40]

J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118. doi: 10.1007/s000280300004.

[41]

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.

[42]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.

show all references

References:
[1]

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.

[2]

P. Bénilan and M. G. Crandall, The continuous dependence on $\phi$ of solutions of $u_t-\Delta\phi(u)=0 $, Indiana Univ. Math. J., 30 (1981), 161-177. doi: 10.1512/iumj.1981.30.30014.

[3]

P. Bénilan and R. Gariepy, Strong solutions in $L^1$ of degenerate parabolic equations, J. Differential Equations, 119 (1995), 473-502. doi: 10.1006/jdeq.1995.1099.

[4]

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Rat. Mech. Anal., 191 (2009), 347-385. doi: 10.1007/s00205-008-0155-z.

[5]

M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, preprint, arXiv:1404.6195, 2014.

[6]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, preprint, arXiv:1311.6997, 2013. doi: 10.1007/s00205-015-0861-2.

[7]

M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018.

[8]

C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.

[9]

H. Brezis and S. Kamin, Sublinear Elliptic Equations in $\mathbb{R}^{N}$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.

[10]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.

[11]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[12]

W. Choi, S. Kim and K.-A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598. doi: 10.1016/j.jfa.2014.02.029.

[13]

A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017.

[14]

A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284. doi: 10.1002/cpa.21408.

[15]

E. Di Benedetto, Continuity of Weak Solutions to a General Porous Medium Equation, Indiana Univ. Math. J., 32 (1983), 83-118. doi: 10.1512/iumj.1983.32.32008.

[16]

G. Di Blasio and B. Volzone, Comparison and regularity results for the fractional Laplacian via symmetrization methods, J. Differential Equations, 253 (2012), 2593-2615. doi: 10.1016/j.jde.2012.07.004.

[17]

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.

[18]

D. Eidus, The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium, J. Differential Equations, 84 (1990), 309-318. doi: 10.1016/0022-0396(90)90081-Y.

[19]

D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830. doi: 10.1090/S0002-9939-1994-1169025-2.

[20]

A. Friedman and S. Kamin, The asymptotic behavior of gas in an $n$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563. doi: 10.2307/1999846.

[21]

G. Grillo, M. Muratori and M. M. Porzio, Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities, Discrete Contin. Dyn. Syst., 33 (2013), 3599-3640. doi: 10.3934/dcds.2013.33.3599.

[22]

G. Grillo, M. Muratori and F. Punzo, Conditions at infinity for the inhomogeneous filtration equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 413-428. doi: 10.1016/j.anihpc.2013.04.002.

[23]

G. Grillo, M. Muratori and F. Punzo, Weighted fractional porous media equations: Existence and uniqueness of weak solutions with measure data, preprint, arXiv:1312.6076, 2013.

[24]

R. G. Iagar and A. Sánchez, Asymptotic behavior for the heat equation in nonhomogeneous media with critical density, Nonlinear Anal., 89 (2013), 24-35. doi: 10.1016/j.na.2013.05.002.

[25]

R. G. Iagar, A. Sánchez, Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density, Nonlinear Anal., 102 (2014), 226-241. doi: 10.1016/j.na.2014.02.016.

[26]

S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 9 (1998), 279-298.

[27]

S. Kamin, G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, Discrete Contin. Dyn. Syst., 26 (2010), 521-549. doi: 10.3934/dcds.2010.26.521.

[28]

S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127. doi: 10.1002/cpa.3160350106.

[29]

S. Nieto and G. Reyes, Asymptotic behavior of the solutions of the inhomogeneous porous medium equation with critical vanishing density, Commun. Pure Appl. Anal., 12 (2013), 1123-1139. doi: 10.3934/cpaa.2013.12.1123.

[30]

M. Pierre, Uniqueness of the solutions of $u_t - \Delta \varphi (u)=0$ with initial datum a measure, Nonlinear Anal., 6 (1982), 175-187. doi: 10.1016/0362-546X(82)90086-4.

[31]

F. Punzo, On the Cauchy problem for nonlinear parabolic equations with variable density, J. Evol. Equ., 9 (2009), 429-447. doi: 10.1007/s00028-009-0018-6.

[32]

F. Punzo and G. Terrone, On the Cauchy problem for a general fractional porous medium equation with variable density, Nonlinear Anal., 98 (2014), 27-47. doi: 10.1016/j.na.2013.12.007.

[33]

F. Punzo and G. Terrone, Well-posedness for the Cauchy problem for a fractional porous medium equation with variable density in one space dimension, Differential Integral Equations, 27 (2014), 461-482.

[34]

F. Punzo and G. Terrone, On a fractional sublinear elliptic equation with a variable coefficient, Appl. Anal., 94 (2015), 800-818. doi: 10.1080/00036811.2014.902053.

[35]

F. Punzo and E. Valdinoci, Uniqueness in weighted Lebesgue spaces for a class of fractional elliptic and parabolic equations, J. Differential Equations, 258 (2015), 555-587. doi: 10.1016/j.jde.2014.09.023.

[36]

V. D. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods, Contemporary Mathematics and Its Applications, 6, Hindawi Publishing Corporation, New York, 2008. doi: 10.1155/9789774540394.

[37]

G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508. doi: 10.3934/cpaa.2009.8.493.

[38]

G. Reyes and J. L. Vázquez, The Cauchy problem for the inhomogeneous porous medium equation, Netw. Heterog. Media, 1 (2006), 337-351. doi: 10.3934/nhm.2006.1.337.

[39]

G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal., 7 (2008), 1275-1294. doi: 10.3934/cpaa.2008.7.1275.

[40]

J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118. doi: 10.1007/s000280300004.

[41]

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.

[42]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.

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