R. A. Adams, Sobolev spaces Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], Pure and Applied Mathematics, vol.65, 1975.

N. D. Alikakos, Bounds of solutions of reaction-diffusion equations, Communications in Partial Differential Equations, vol.11, issue.8, pp.827-868, 1979.
DOI : 10.1002/cpa.3160130308

L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, 2000.

D. Andreucci, G. R. Cirmi, S. Leonardi, and A. Tedeev, Large Time Behavior of Solutions to the Neumann Problem for a Quasilinear Second Order Degenerate Parabolic Equation in Domains with Noncompact Boundary, Journal of Differential Equations, vol.174, issue.2, pp.253-288, 2001.
DOI : 10.1006/jdeq.2000.3948

D. Andreucci and A. Tedeev, Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity, Adv. Differential Equations, vol.5, pp.7-9, 2000.

G. Anzellotti and S. Baldo, Asymptotic development by ?-convergence, Applied Mathematics & Optimization, vol.14, issue.2, pp.105-123, 1993.
DOI : 10.1007/BF01195977

D. Aronson, M. G. Crandall, and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Analysis: Theory, Methods & Applications, vol.6, issue.10, pp.1001-1022, 1982.
DOI : 10.1016/0362-546X(82)90072-4

D. G. Aronson, C. , and L. A. , The initial trace of a solution of the porous medium equation, Transactions of the American Mathematical Society, vol.280, issue.1, pp.1-351, 1983.
DOI : 10.1090/S0002-9947-1983-0712265-1

I. Athanasopoulos, C. , and L. A. , Continuity of the temperature in boundary heat control problems, Advances in Mathematics, vol.224, issue.1, pp.293-315, 2010.
DOI : 10.1016/j.aim.2009.11.010

T. Aubin, Probl??mes isop??rim??triques et espaces de Sobolev, Journal of Differential Geometry, vol.11, issue.4, pp.573-598, 1976.
DOI : 10.4310/jdg/1214433725

D. Bakry, F. Barthe, P. Cattiaux, and A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case, Electron. Commun. Probab, pp.13-60, 2008.

D. Bakry, T. Coulhon, M. Ledoux, and L. Saloff-coste, Sobolev inequalities in disguise, Indiana University Mathematics Journal, vol.44, issue.4, pp.1033-1074, 1995.
DOI : 10.1512/iumj.1995.44.2019

D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators, of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 2014.
DOI : 10.1007/978-3-319-00227-9
URL : https://hal.archives-ouvertes.fr/hal-00929960

B. Barrios, I. Peral, F. Soria, and E. Valdinoci, A Widder???s Type Theorem for the Heat Equation with Nonlocal Diffusion, Archive for Rational Mechanics and Analysis, vol.55, issue.1, pp.629-650, 2014.
DOI : 10.1007/s00205-014-0733-1

P. Bénilan and R. Gariepy, Strong Solutions in L1 of Degenerate Parabolic Equations, Journal of Differential Equations, vol.119, issue.2, pp.473-502, 1995.
DOI : 10.1006/jdeq.1995.1099

E. Berchio, A. Ferrero, and G. Grillo, Stability and qualitative properties of radial solutions of the Lane???Emden???Fowler equation on Riemannian models, Journal de Math??matiques Pures et Appliqu??es, vol.102, issue.1, pp.1-35, 2014.
DOI : 10.1016/j.matpur.2013.10.012

J. G. Berryman and C. J. Holland, Stability of the separable solution for fast diffusion, Archive for Rational Mechanics and Analysis, vol.74, issue.4, pp.379-388, 1980.
DOI : 10.1007/BF00249681

A. L. Besse, Einstein manifolds, Classics in Mathematics, 2008.
DOI : 10.1007/978-3-540-74311-8

M. Bhakta and K. Sandeep, Poincar?????Sobolev equations in the hyperbolic space, Calculus of Variations and Partial Differential Equations, vol.18, issue.9, pp.1-2, 2012.
DOI : 10.1007/s00526-011-0433-8

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, and J. L. Vázquez, Hardy???Poincar?? inequalities and applications to nonlinear diffusions, Comptes Rendus Mathematique, vol.344, issue.7, pp.431-436, 2007.
DOI : 10.1016/j.crma.2007.01.011

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, and J. L. Vázquez, Asymptotics of the Fast Diffusion Equation via Entropy Estimates, Archive for Rational Mechanics and Analysis, vol.1, issue.4, pp.347-385, 2009.
DOI : 10.1007/s00205-008-0155-z
URL : https://hal.archives-ouvertes.fr/hal-00142404

M. Bonforte, J. Dolbeault, G. Grillo, and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. USA, pp.38-16459, 2010.
DOI : 10.1073/pnas.1003972107
URL : https://hal.archives-ouvertes.fr/hal-00404718

M. Bonforte, J. Dolbeault, M. Muratori, and B. Nazaret, Weighted fast diffusion equations: sharp rates of convergence without symmetry

M. Bonforte, F. Gazzola, G. Grillo, and J. L. Vázquez, Classification of radial solutions to the Emden???Fowler equation on the hyperbolic space, Calculus of Variations and Partial Differential Equations, vol.142, issue.5, pp.1-2, 2013.
DOI : 10.1007/s00526-011-0486-8

M. Bonforte and G. Grillo, Asymptotics of the porous media equation via Sobolev inequalities, Journal of Functional Analysis, vol.225, issue.1, pp.33-62, 2005.
DOI : 10.1016/j.jfa.2005.03.011

M. Bonforte and G. Grillo, Ultracontractive Bounds for Nonlinear Evolution Equations Governed by the Subcritical p-Laplacian, Trends in partial differential equations of mathematical physics, pp.15-26, 2005.
DOI : 10.1007/3-7643-7317-2_2

M. Bonforte, G. Grillo, and J. L. Vázquez, Fast diffusion flow on manifolds of nonpositive curvature, Journal of Evolution Equations, vol.8, issue.1, pp.99-128, 2008.
DOI : 10.1007/s00028-007-0345-4

M. Bonforte, G. Grillo, and J. L. Vázquez, Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold, Archive for Rational Mechanics and Analysis, vol.12, issue.1, pp.631-680, 2010.
DOI : 10.1007/s00205-009-0252-7

M. Bonforte, G. Grillo, and J. L. Vázquez, Behaviour near extinction for the Fast Diffusion Equation on bounded domains, Journal de Math??matiques Pures et Appliqu??es, vol.97, issue.1, pp.1-38, 2012.
DOI : 10.1016/j.matpur.2011.03.002

M. Bonforte, Y. Sire, and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete and Continuous Dynamical Systems, vol.35, issue.12, 2014.
DOI : 10.3934/dcds.2015.35.5725
URL : https://hal.archives-ouvertes.fr/hal-01005326

M. Bonforte and J. L. Vázquez, A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains, Archive for Rational Mechanics and Analysis, vol.182, issue.2, 2013.
DOI : 10.1007/s00205-015-0861-2

M. Bonforte and J. L. Vázquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation, Journal of Functional Analysis, vol.240, issue.2, pp.399-428, 2006.
DOI : 10.1016/j.jfa.2006.07.009

M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Advances in Mathematics, vol.250, pp.242-284, 2014.
DOI : 10.1016/j.aim.2013.09.018

C. Brändle, E. Colorado, A. De-pablo, and U. Sánchez, A concave???convex elliptic problem involving the fractional Laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.143, issue.01, pp.39-71, 2013.
DOI : 10.1017/S0308210511000175

H. Brézis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl, vol.62, issue.1, pp.73-97, 1983.

H. Brézis and S. Kamin, Sublinear elliptic equations in ???n, Manuscripta Mathematica, vol.14, issue.1, pp.87-106, 1992.
DOI : 10.1007/BF02567660

S. M. Buckley and P. Koskela, New Poincaré inequalities from old, Ann. Acad. Sci. Fenn. Math, vol.23, issue.1, pp.251-260, 1998.

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.31, issue.1, pp.23-53, 2014.
DOI : 10.1016/j.anihpc.2013.02.001

L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math, vol.53, issue.3, pp.259-275, 1984.

L. Caffarelli and L. Silvestre, An Extension Problem Related to the Fractional Laplacian, Communications in Partial Differential Equations, vol.32, issue.8, pp.1245-1260, 2007.
DOI : 10.1080/00036818308839425

J. A. Carrillo, D. Francesco, M. Toscani, and G. , Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer, pp.353-363, 2007.

J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani, and A. Unterreiter, Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities, Monatshefte f???r Mathematik, vol.133, issue.1, pp.1-1, 2001.
DOI : 10.1007/s006050170032

J. A. Carrillo, C. Lederman, P. A. Markowich, and G. Toscani, Poincar?? inequalities for linearizations of very fast diffusion equations, Nonlinearity, vol.15, issue.3, pp.565-580, 2002.
DOI : 10.1088/0951-7715/15/3/303

J. A. Carrillo and G. Toscani, Asymptotic L^1-decay of solutions of the porous medium equation to self-similarity, Indiana University Mathematics Journal, vol.49, issue.1, pp.113-142, 2000.
DOI : 10.1512/iumj.2000.49.1756

F. Catrina, W. , and Z. , On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Communications on Pure and Applied Mathematics, vol.24, issue.2, pp.229-258, 2001.
DOI : 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I

E. Chasseigne and J. L. Vázquez, Theory of Extended Solutions??for Fast-Diffusion Equations??in Optimal Classes of Data.??Radiation from Singularities, Archive for Rational Mechanics and Analysis, vol.164, issue.2, pp.133-187, 2002.
DOI : 10.1007/s00205-002-0210-0

E. Chasseigne and J. L. Vázquez, Extended solutions for general fast diffusion equations with optimal measure data, Adv. Differential Equations, vol.11, issue.6, pp.627-646, 2006.

H. Chen, L. Véron, W. , and Y. , Fractional heat equations involving initial measure data and subcritical absorption, Preprint, 2014.

Y. Z. Chen and E. Dibenedetto, On the local behavior of solutions of singular parabolic equations, Arch. Rational Mech. Anal, vol.103, issue.4, pp.319-345, 1988.

W. Choi, S. Kim, L. , and K. , Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, Journal of Functional Analysis, vol.266, issue.11, pp.11-6531, 2014.
DOI : 10.1016/j.jfa.2014.02.029

K. S. Chou, C. , and C. W. , On the Best Constant for a Weighted Sobolev-Hardy Inequality, Journal of the London Mathematical Society, vol.2, issue.1, pp.137-151, 1993.
DOI : 10.1112/jlms/s2-48.1.137

S. Chua and R. L. Wheeden, Sharp conditions for 1-dimensional weighted Poincare inequalities, Indiana University Mathematics Journal, vol.49, issue.1, pp.143-175, 2000.
DOI : 10.1512/iumj.2000.49.1754

S. Chua and R. L. Wheeden, Weighted Poincar?? inequalities on convex domains, Mathematical Research Letters, vol.17, issue.5, pp.993-1011, 2010.
DOI : 10.4310/MRL.2010.v17.n5.a15

B. E. Dahlberg and C. Kenig, Non-negative solutions of theo porous medium eqation, Communications in Partial Differential Equations, vol.55, issue.5, pp.409-437, 1984.
DOI : 10.1080/03605308408820336

D. 'ancona, P. Luca-', and R. , Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities with angular integrability, J. Math. Anal. Appl, vol.388, issue.2, pp.1061-1079, 2012.

P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion, Journal f??r die reine und angewandte Mathematik (Crelles Journal), vol.2008, issue.622, pp.95-119, 2008.
DOI : 10.1515/CRELLE.2008.066

E. B. Davies, Heat kernels and spectral theory of Cambridge Tracts in Mathematics, 1989.

A. De-pablo, F. Quirós, A. Rodríguez, and J. L. Vázquez, A fractional porous medium equation, Advances in Mathematics, vol.226, issue.2, pp.1378-1409, 2011.
DOI : 10.1016/j.aim.2010.07.017

A. De-pablo, F. Quirós, A. Rodríguez, and J. L. Vázquez, A General Fractional Porous Medium Equation, Communications on Pure and Applied Mathematics, vol.63, issue.2, pp.9-1242, 2012.
DOI : 10.1002/cpa.21408

D. Pino, M. Dolbeault, and J. , Best constants for Gagliardo???Nirenberg inequalities and applications to nonlinear diffusions, Journal de Math??matiques Pures et Appliqu??es, vol.81, issue.9, pp.847-875, 2002.
DOI : 10.1016/S0021-7824(02)01266-7

D. Pino, M. Dolbeault, J. Filippas, S. Tertikas, and A. , A logarithmic Hardy inequality, Journal of Functional Analysis, vol.259, issue.8, pp.2045-2072, 2010.
DOI : 10.1016/j.jfa.2010.06.005
URL : https://hal.archives-ouvertes.fr/hal-00438199

J. Denzler, H. Koch, and R. J. Mccann, Higher-order time asymptotics of fast diffusion in Euclidean space: A dynamical systems approach, Memoirs of the American Mathematical Society, vol.234, issue.1101, 2012.
DOI : 10.1090/memo/1101

J. Denzler and R. J. Mccann, Phase transitions and symmetry breaking in singular diffusion, Proc. Natl. Acad. Sci. USA, pp.6922-6925, 2003.
DOI : 10.1073/pnas.1231896100

J. Denzler and R. J. Mccann, Fast Diffusion to Self-Similarity: Complete Spectrum, Long-Time Asymptotics, and Numerology, Archive for Rational Mechanics and Analysis, vol.3, issue.3, pp.3-301, 2005.
DOI : 10.1007/s00205-004-0336-3

B. Devyver, M. Fraas, and Y. Pinchover, Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon, Journal of Functional Analysis, vol.266, issue.7, pp.7-4422, 2014.
DOI : 10.1016/j.jfa.2014.01.017

D. Blasio, G. Volzone, and B. , Comparison and regularity results for the fractional Laplacian via symmetrization methods, Journal of Differential Equations, vol.253, issue.9, pp.2593-2615, 2012.
DOI : 10.1016/j.jde.2012.07.004

D. Nezza, E. Palatucci, G. Valdinoci, and E. , Hitchhiker??s guide to the fractional Sobolev spaces, Bulletin des Sciences Math??matiques, vol.136, issue.5, pp.521-573, 2012.
DOI : 10.1016/j.bulsci.2011.12.004

E. Dibenedetto, A boundary modulus of continuity for a class of singular parabolic equations, Journal of Differential Equations, vol.63, issue.3, pp.418-447, 1986.
DOI : 10.1016/0022-0396(86)90064-1

E. Dibenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math, vol.357, pp.1-22, 1985.

J. Dolbeault, E. , and M. J. , About Existence, Symmetry and Symmetry Breaking for Extremal Functions of Some Interpolation Functional Inequalities, of Abel Symposia, pp.117-130
DOI : 10.1007/978-3-642-25361-4_6
URL : https://hal.archives-ouvertes.fr/hal-00539019

J. Dolbeault, E. , and M. J. , Extremal functions for Caffarelli???Kohn???Nirenberg and logarithmic Hardy inequalities, Proc. Roy. Soc. Edinburgh Sect. A 142, pp.745-767, 2012.
DOI : 10.1017/S0308210510001101
URL : https://hal.archives-ouvertes.fr/hal-00496936

J. Dolbeault, E. , and M. J. , A scenario for symmetry breaking in Caffarelli???Kohn???Nirenberg inequalities, Journal of Numerical Mathematics, vol.20, issue.3-4, pp.3-4, 2012.
DOI : 10.1515/jnum-2012-0012
URL : https://hal.archives-ouvertes.fr/hal-00695542

J. Dolbeault, E. , and M. J. , Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations, Nonlinearity, vol.27, issue.3, pp.435-465, 2014.
DOI : 10.1088/0951-7715/27/3/435
URL : https://hal.archives-ouvertes.fr/hal-00812996

J. Dolbeault, M. J. Esteban, and M. Loss, Symmetry of extremals of functional inequalities via spectral estimates for linear operators, Journal of Mathematical Physics, vol.53, issue.9, pp.95204-95222, 2012.
DOI : 10.1063/1.4704911
URL : https://hal.archives-ouvertes.fr/hal-00626739

J. Dolbeault, M. J. Esteban, M. Loss, and G. Tarantello, Abstract, Advanced Nonlinear Studies, vol.9, issue.4, pp.713-726, 2009.
DOI : 10.1515/ans-2009-0407

J. Dolbeault, M. J. Esteban, and G. Tarantello, The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, Ann. Sc. Norm. Super. Pisa Cl. Sci, vol.7, issue.5 2, pp.313-341, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00139062

J. Dolbeault, M. J. Esteban, G. Tarantello, and A. Tertikas, Radial symmetry and symmetry breaking for some interpolation inequalities, Calculus of Variations and Partial Differential Equations, vol.237, issue.2, pp.3-4, 2011.
DOI : 10.1007/s00526-011-0394-y
URL : https://hal.archives-ouvertes.fr/hal-00516710

J. Dolbeault, I. Gentil, A. Guillin, W. , and F. , L q -functional inequalities and weighted porous media equations. Potential Anal, pp.1-35, 2008.

J. Dolbeault, B. Nazaret, and G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations, Communications in Mathematical Sciences, vol.6, issue.2, pp.477-494, 2008.
DOI : 10.4310/CMS.2008.v6.n2.a10
URL : https://hal.archives-ouvertes.fr/hal-00196935

J. Dolbeault and G. Toscani, Best matching Barenblatt profiles are delayed, Journal of Physics A: Mathematical and Theoretical, vol.48, issue.6, 2014.
DOI : 10.1088/1751-8113/48/6/065206
URL : https://hal.archives-ouvertes.fr/hal-01058838

J. Dolbeault and G. Toscani, Fast diffusion equations: Matching large time asymptotics by relative entropy methods, Kinetic and Related Models, vol.4, issue.3, pp.701-716, 2011.
DOI : 10.3934/krm.2011.4.701
URL : https://hal.archives-ouvertes.fr/hal-00482898

J. Dolbeault and G. Toscani, Improved interpolation inequalities, relative entropy and fast diffusion equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.30, issue.5, pp.917-934, 2013.
DOI : 10.1016/j.anihpc.2012.12.004
URL : https://hal.archives-ouvertes.fr/hal-00634852

I. Drelichman and R. G. Durán, Improved Poincar?? inequalities with weights, Journal of Mathematical Analysis and Applications, vol.347, issue.1, pp.286-293, 2008.
DOI : 10.1016/j.jmaa.2008.06.005

B. Dyda and M. Kassmann, On weighted Poincar?? inequalities, Annales Academiae Scientiarum Fennicae Mathematica, vol.38, issue.2, pp.721-726, 2013.
DOI : 10.5186/aasfm.2013.3834

D. Eidus, The Cauchy problem for the non-linear filtration equation in an inhomogeneous medium, Journal of Differential Equations, vol.84, issue.2, pp.309-318, 1990.
DOI : 10.1016/0022-0396(90)90081-Y

D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer, pp.825-830, 1994.
DOI : 10.1090/S0002-9939-1994-1169025-2

E. Fabes, M. Fukushima, L. Gross, C. Kenig, M. Röckner et al., Dirichlet forms Lectures given at the First C.I.M.E. Session held in Varenna, Lecture Notes in Mathematics, vol.1563, 1992.

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, Journal of Dynamics and Differential Equations, vol.12, issue.3, pp.647-673, 2000.
DOI : 10.1023/A:1026467729263

V. Felli and M. Schneider, Perturbation results of critical elliptic equations of Caffarelli???Kohn???Nirenberg type, Journal of Differential Equations, vol.191, issue.1, pp.121-142, 2003.
DOI : 10.1016/S0022-0396(02)00085-2

S. Filippas, L. Moschini, and A. Tertikas, Sharp Two???Sided Heat Kernel Estimates for Critical Schr??dinger Operators on Bounded Domains, Communications in Mathematical Physics, vol.182, issue.3???4, pp.237-281, 2007.
DOI : 10.1007/s00220-007-0253-z

R. L. Frank, E. H. Lieb, and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schr??dinger operators, Journal of the American Mathematical Society, vol.21, issue.4, pp.925-950, 2008.
DOI : 10.1090/S0894-0347-07-00582-6

A. Friedman, On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations, J. Math. Mech, vol.7, pp.43-59, 1958.

A. Friedman, Partial differential equations of parabolic type, N.J, 1964.

A. Friedman and S. Kamin, The asymptotic behavior of gas in an n-dimensional porous medium, Trans. Amer. Math. Soc, vol.262, issue.2, pp.551-563, 1980.

E. Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat, vol.7, pp.102-137, 1958.

E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat, vol.8, pp.24-51, 1959.

B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in R n, Mathematical analysis and applications, Part A, pp.369-402, 1981.

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, 2001.

R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol.699, 1979.
DOI : 10.1007/BFb0063413

G. Grillo, On the equivalence between p-Poincar?? inequalities and <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mi mathvariant="normal">L</mml:mi><mml:mi>r</mml:mi></mml:msup></mml:math>???<mml:math altimg="si2.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mi mathvariant="normal">L</mml:mi><mml:mi>q</mml:mi></mml:msup></mml:math> regularization and decay estimates of certain nonlinear evolutions, Journal of Differential Equations, vol.249, issue.10, pp.2561-2576, 2010.
DOI : 10.1016/j.jde.2010.05.022

L. Gross, Logarithmic Sobolev Inequalities, American Journal of Mathematics, vol.97, issue.4, pp.1061-1083, 1975.
DOI : 10.2307/2373688

M. Guedda, D. Hilhorst, and M. A. Peletier, Disappearing interfaces in nonlinear diffusion, Adv. Math. Sci. Appl, vol.7, issue.2, pp.695-710, 1997.

M. E. Gurtin and R. C. Maccamy, On the diffusion of biological populations, Mathematical Biosciences, vol.33, issue.1-2, pp.1-2, 1977.
DOI : 10.1016/0025-5564(77)90062-1

T. Horiuchi, Best constant in weighted sobolev inequality with weights being powers of distance from the origin, Journal of Inequalities and Applications, vol.1997, issue.3, pp.275-292, 1997.
DOI : 10.1155/S1025583497000180

R. Hurri, The weighted Poincar?? inequalities., MATHEMATICA SCANDINAVICA, vol.67, issue.1, pp.145-160, 1990.
DOI : 10.7146/math.scand.a-12325

R. G. Iagar and A. Sánchez, Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density, Nonlinear Analysis: Theory, Methods & Applications, vol.102, pp.226-241, 2014.
DOI : 10.1016/j.na.2014.02.016

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, vol.9, issue.4, pp.279-298, 1998.

S. Kamin, M. A. Pozio, and A. Tesei, Admissible conditions for parabolic equations degenerating at infinity, St. Petersburg Mathematical Journal, vol.19, issue.02, pp.105-121, 2007.
DOI : 10.1090/S1061-0022-08-00996-5

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, vol.26, issue.2, pp.521-549, 2010.

S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Communications on Pure and Applied Mathematics, vol.3, issue.6, pp.831-852, 1981.
DOI : 10.1002/cpa.3160340605

S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math, vol.35, issue.1, pp.113-127, 1982.

R. Kersner and S. Kamin, Disappearance of interfaces in finite time, Meccanica, vol.28, pp.117-120, 1993.

R. Kersner, G. Reyes, and A. Tesei, On a class of parabolic equations with variable density and absorption, Adv. Differential Equations, vol.7, issue.2, pp.155-176, 2002.

A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin, vol.25, issue.3, pp.537-554, 1984.

O. A. Lady?enskaja, V. A. Solonnikov, and N. N. Ural-'ceva, Linear and quasilinear equations of parabolic type. Translated from the Russian by S, Smith. Translations of Mathematical Monographs, vol.23, 1968.

N. S. Landkof, Foundations of modern potential theory, 1972.
DOI : 10.1007/978-3-642-65183-0

C. Lederman and P. A. Markowich, On Fast-Diffusion Equations with Infinite Equilibrium Entropy and Finite Equilibrium Mass, Communications in Partial Differential Equations, vol.71, issue.9, pp.1-2, 2003.
DOI : 10.1081/PDE-120019384

K. Lee, A. Petrosyan, and J. L. Vázquez, Large-time geometric properties of solutions of the evolution p-Laplacian equation, Journal of Differential Equations, vol.229, issue.2, pp.389-411, 2006.
DOI : 10.1016/j.jde.2005.07.028

K. Lee and J. L. Vázquez, Geometrical properties of solutions of the Porous Medium Equation for large times, Indiana University Mathematics Journal, vol.52, issue.4, pp.991-1016, 2003.
DOI : 10.1512/iumj.2003.52.2200

G. M. Lieberman, Second order parabolic differential equations, 1996.
DOI : 10.1142/3302

C. Lin, W. , and Z. , Symmetry of extremal functions for the Caffarrelli-Kohn-Nirenberg inequalities, Proc. Amer, pp.1685-1691, 2004.

P. Lions, The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1. * *Mp denotes the Marcinkiewicz space or weak Lp space, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.1, issue.2, pp.109-145, 1984.
DOI : 10.1016/S0294-1449(16)30428-0

P. Lions, The concentration-compactness principle in the Calculus of Variations. The Locally compact case, part 2, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.1, issue.4, pp.223-283, 1984.
DOI : 10.1016/S0294-1449(16)30422-X

G. Mancini and K. Sandeep, On a semilinear elliptic equation in H n, Ann. Sc. Norm. Super. Pisa Cl. Sci, vol.7, issue.5 4, pp.635-671, 2008.

M. Marcus and L. Véron, Capacitary estimates of solutions of semilinear parabolic equations, Calculus of Variations and Partial Differential Equations, vol.24, issue.9, pp.1-2, 2013.
DOI : 10.1007/s00526-012-0545-9
URL : https://hal.archives-ouvertes.fr/hal-00282315

V. G. Maz-'ja, Sobolev spaces, 1985.

R. J. Mccann and D. Slep?ev, Second-order asymptotics for the fast-diffusion equation, International Mathematics Research Notices, p.22, 2006.
DOI : 10.1155/IMRN/2006/24947

G. Mingione, Nonlinear Measure Data Problems, Milan Journal of Mathematics, vol.23, issue.3, pp.429-496, 2011.
DOI : 10.1007/s00032-011-0168-1

J. Moser, On Harnack's theorem for elliptic differential equations, Communications on Pure and Applied Mathematics, vol.13, issue.3, pp.577-591, 1961.
DOI : 10.1002/cpa.3160140329

J. Moser, A harnack inequality for parabolic differential equations, Communications on Pure and Applied Mathematics, vol.55, issue.1, pp.101-134, 1964.
DOI : 10.1002/cpa.3160170106

J. Moser, Correetion to ???A Harnack Inequality for Parabolic Differential Equations???, Communications on Pure and Applied Mathematics, vol.20, issue.1, pp.231-236, 1967.
DOI : 10.1002/cpa.3160200107

J. Moser, On a pointwise estimate for parabolic differential equations, Communications on Pure and Applied Mathematics, vol.20, issue.5, pp.727-740, 1971.
DOI : 10.1002/cpa.3160240507

B. Muckenhoupt, Hardy's inequality with weights Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, Studia Math, vol.44, pp.31-38, 1972.

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Transactions of the American Mathematical Society, vol.165, pp.207-226, 1972.
DOI : 10.1090/S0002-9947-1972-0293384-6

M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, Soil Science, vol.46, issue.2, 1937.
DOI : 10.1097/00010694-193808000-00008

S. Nieto, R. , and G. , Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density, Communications on Pure and Applied Analysis, vol.12, issue.2, pp.1123-1139, 2013.
DOI : 10.3934/cpaa.2013.12.1123

L. Nirenberg, On Elliptic Partial Differential Equations, Ann. Scuola Norm. Sup. Pisa, vol.13, issue.3, pp.115-162, 1959.
DOI : 10.1007/978-3-642-10926-3_1

O. A. Ole?-inik, On the equations of unsteady filtration, Dokl. Akad. Nauk SSSR (N.S.), vol.113, pp.1210-1213, 1957.

O. A. Ole?-inik, A. S. Kala?inkov, and Y. ??ou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat, pp.22-667, 1958.

B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, vol.219, 1990.

A. Pazy, Semigroups of linear operators and applications to partial differential equations, of Applied Mathematical Sciences, 1983.
DOI : 10.1007/978-1-4612-5561-1

M. A. Peletier, A supersolution for the porous media equation with nonuniform density, Applied Mathematics Letters, vol.7, issue.3, pp.29-32, 1994.
DOI : 10.1016/0893-9659(94)90108-2

M. Pierre, Uniqueness of the solutions of ut???????(u) = 0 with initial datum a measure, Nonlinear Analysis: Theory, Methods & Applications, vol.6, issue.2, pp.175-187, 1982.
DOI : 10.1016/0362-546X(82)90086-4

M. M. Porzio, On decay estimates, Journal of Evolution Equations, vol.18, issue.3, pp.561-591, 2009.
DOI : 10.1007/s00028-009-0024-8

M. M. Porzio, F. Smarrazzo, and A. Tesei, Radon Measure-Valued Solutions for a Class of Quasilinear Parabolic Equations, Archive for Rational Mechanics and Analysis, vol.143, issue.3, pp.3-713, 2013.
DOI : 10.1007/s00205-013-0666-0

M. M. Porzio and V. Vespri, Holder Estimates for Local Solutions of Some Doubly Nonlinear Degenerate Parabolic Equations, Journal of Differential Equations, vol.103, issue.1, pp.146-178, 1993.
DOI : 10.1006/jdeq.1993.1045

P. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana University Mathematics Journal, vol.47, issue.2, pp.501-528, 1998.
DOI : 10.1512/iumj.1998.47.1517

F. Punzo, On the Cauchy problem for nonlinear parabolic equations with variable density, Journal of Evolution Equations, vol.2, issue.3, pp.429-447, 2009.
DOI : 10.1007/s00028-009-0018-6

F. Punzo, Uniqueness and non-uniqueness of bounded solutions to singular nonlinear parabolic equations . Nonlinear Anal, pp.3020-3029, 2009.

F. Punzo, On support of solutions to singular nonlinear parabolic equations in bounded domains. Interfaces Free Bound, pp.397-410, 2011.

F. Punzo, Uniqueness and non-uniqueness of solutions to quasilinear parabolic equations with a singular coefficient on weighted Riemannian manifolds, Asymptot. Anal, vol.79, pp.3-4, 2012.

F. Punzo, Uniqueness and support properties of solutions to singular quasilinear parabolic equations on surfaces of revolution, Annali di Matematica Pura ed Applicata, vol.2, issue.2, pp.311-338, 2012.
DOI : 10.1007/s10231-011-0185-2

F. Punzo and G. Terrone, On a fractional sublinear elliptic equation with a variable coefficient, Applicable Analysis, vol.5, issue.4
DOI : 10.1016/j.jde.2012.07.004

F. Punzo and G. Terrone, On the Cauchy problem for a general fractional porous medium equation with variable density. Nonlinear Anal, pp.98-125, 2014.

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, vol.27, pp.5-6, 2014.

F. Punzo and E. Valdinoci, Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations, Journal of Differential Equations, vol.258, issue.2, pp.555-587, 2015.
DOI : 10.1016/j.jde.2014.09.023

G. Reyes and J. L. Vázquez, The Cauchy problem for the inhomogeneous porous medium equation, Netw. Heterog. Media, vol.1, issue.2, pp.337-351, 2006.

G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal, vol.7, issue.6, pp.1275-1294, 2008.

G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal, vol.8, issue.2, pp.493-508, 2009.

D. Smets, W. , and M. , Partial symmetry and asymptotic behavior for some elliptic variational problems, Calculus of Variations and Partial Differential Equations, vol.18, issue.1, pp.57-75, 2003.
DOI : 10.1007/s00526-002-0180-y

D. Stan and J. L. Vázquez, Asymptotic behaviour of the doubly nonlinear diffusion equation on bounded domains, Nonlinear Analysis: Theory, Methods & Applications, vol.77, pp.1-32, 2013.
DOI : 10.1016/j.na.2012.08.011

G. Talenti, Best constant in Sobolev inequality, Annali di Matematica Pura ed Applicata, Series 4, vol.110, issue.1, pp.353-372, 1976.
DOI : 10.1007/BF02418013
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.615.4193

A. F. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations, Applicable Analysis, vol.1, issue.6, pp.755-782, 2007.
DOI : 10.1016/0022-247X(88)90053-4

J. L. Vázquez, Fundamental solution and long time behaviour of the porous medium equation in hyperbolic space, Preprint, 2014.

J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, Journal of Evolution Equations, vol.3, issue.1, pp.67-118, 2003.
DOI : 10.1007/s000280300004

J. L. Vázquez, The Dirichlet problem for the porous medium equation in bounded domains Asymptotic behavior, Monatsh. Math, vol.142, pp.1-2, 2004.

J. L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations, of Oxford Lecture Series in Mathematics and its Applications, 2006.
DOI : 10.1093/acprof:oso/9780199202973.001.0001

J. L. Vázquez, The porous medium equation. Oxford Mathematical Monographs, 2007.

J. L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type, Journal of the European Mathematical Society, vol.16, issue.4, pp.769-803, 2014.
DOI : 10.4171/JEMS/446

J. L. Vázquez, A. De-pablo, F. Quirós, and A. Rodríguez, Classical solutions and higher regularity for nonlinear fractional diffusion equations, Preprint, 2013.

F. Wang, ORLICZ???POINCAR?? INEQUALITIES, Proc. Edinb. Math. Soc. (2) 51, pp.529-543, 2008.
DOI : 10.1017/S0013091506000526