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

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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

Abstract
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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 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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