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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
H. Brezis and S. Kamin, Sublinear Elliptic Equations in $\mathbbR^n$,, Manuscripta Math., 74 (1992), 87.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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, (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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[42] |
J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs, (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.
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.
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.
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.
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, (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. |
[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. |
[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. |
[9] |
H. Brezis and S. Kamin, Sublinear Elliptic Equations in $\mathbbR^n$,, Manuscripta Math., 74 (1992), 87.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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, (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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[42] |
J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs, (2007).
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