American Institute of Mathematical Sciences

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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

Gabriele Grillo 1, , Matteo Muratori 1, and  Fabio Punzo 2,

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.
Keywords: nonlinear diffusion equations, fractional Laplacian, Porous medium equation, separable variable solutions., asymptotics of solutions, Barenblatt solution.
Mathematics Subject Classification: Primary: 35B40, 35K55; Secondary: 35C06, 35K6.
Citation: Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.aim.2009.11.010.  Google Scholar

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

[3]

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

[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.  doi: 10.1007/s00205-008-0155-z.  Google Scholar

[5]

M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains,, preprint, (2014).   Google Scholar

[6]

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

[7]

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

[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.  doi: 10.1017/S0308210511000175.  Google Scholar

[9]

H. Brezis and S. Kamin, Sublinear Elliptic Equations in $\mathbbR^n$,, Manuscripta Math., 74 (1992), 87.  doi: 10.1007/BF02567660.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[11]

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

[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.  doi: 10.1016/j.jfa.2014.02.029.  Google Scholar

[13]

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

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  doi: 10.3934/dcds.2013.33.3599.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2013.04.002.  Google Scholar

[23]

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

[24]

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

[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.  doi: 10.1016/j.na.2014.02.016.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2010.26.521.  Google Scholar

[28]

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

[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.  doi: 10.3934/cpaa.2013.12.1123.  Google Scholar

[30]

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

[31]

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

[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.  doi: 10.1016/j.na.2013.12.007.  Google Scholar

[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.   Google Scholar

[34]

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

[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.  doi: 10.1016/j.jde.2014.09.023.  Google Scholar

[36]

V. D. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods,, Contemporary Mathematics and Its Applications, (2008).  doi: 10.1155/9789774540394.  Google Scholar

[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.  doi: 10.3934/cpaa.2009.8.493.  Google Scholar

[38]

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

[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.  doi: 10.3934/cpaa.2008.7.1275.  Google Scholar

[40]

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

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

[42]

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

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.  doi: 10.1016/j.aim.2009.11.010.  Google Scholar

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

[3]

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

[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.  doi: 10.1007/s00205-008-0155-z.  Google Scholar

[5]

M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains,, preprint, (2014).   Google Scholar

[6]

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

[7]

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

[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.  doi: 10.1017/S0308210511000175.  Google Scholar

[9]

H. Brezis and S. Kamin, Sublinear Elliptic Equations in $\mathbbR^n$,, Manuscripta Math., 74 (1992), 87.  doi: 10.1007/BF02567660.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[11]

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

[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.  doi: 10.1016/j.jfa.2014.02.029.  Google Scholar

[13]

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

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  doi: 10.3934/dcds.2013.33.3599.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2013.04.002.  Google Scholar

[23]

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

[24]

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

[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.  doi: 10.1016/j.na.2014.02.016.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2010.26.521.  Google Scholar

[28]

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

[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.  doi: 10.3934/cpaa.2013.12.1123.  Google Scholar

[30]

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

[31]

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

[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.  doi: 10.1016/j.na.2013.12.007.  Google Scholar

[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.   Google Scholar

[34]

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

[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.  doi: 10.1016/j.jde.2014.09.023.  Google Scholar

[36]

V. D. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods,, Contemporary Mathematics and Its Applications, (2008).  doi: 10.1155/9789774540394.  Google Scholar

[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.  doi: 10.3934/cpaa.2009.8.493.  Google Scholar

[38]

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

[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.  doi: 10.3934/cpaa.2008.7.1275.  Google Scholar

[40]

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

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

[42]

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

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