Citation: |
[1] |
R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. |
[2] |
N. D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. Math. J., 30 (1981), 749-785.doi: 10.1512/iumj.1981.30.30056. |
[3] |
D. Andreucci, G. R. Cirmi, S. Leonardi and A. F. Tedeev, Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary, J. Differential Equations, 174 (2001), 253-288.doi: 10.1006/jdeq.2000.3948. |
[4] |
D. Andreucci and A. F. Tedeev, Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity, Adv. Differential Equations, 5 (2000), 833-860. |
[5] |
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, Elect. Comm. Prob., 13 (2008), 60-66.doi: 10.1214/ECP.v13-1352. |
[6] |
D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J., 44 (1995), 1033-1074.doi: 10.1512/iumj.1995.44.2019. |
[7] |
A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Hardy-Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris, 344 (2007), 431-436.doi: 10.1016/j.crma.2007.01.011. |
[8] |
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, 107 (2010), 16459-16464.doi: 10.1073/pnas.1003972107. |
[9] |
M. Bonforte and G. Grillo, Asymptotics of the porous media equation via Sobolev inequalities, J. Funct. Anal., 225 (2005), 33-62.doi: 10.1016/j.jfa.2005.03.011. |
[10] |
M. Bonforte and G. Grillo, Ultracontractive bounds for nonlinear evolution equations governed by the subcritical $p$-Laplacian, in "Trends in Partial Differential Equations of Mathematical Physics," Progr. Nonlinear Differential Equations Appl., 61, Birkhäuser, Basel, (2005), 15-26.doi: 10.1007/3-7643-7317-2_2. |
[11] |
M. Bonforte, G. Grillo and J. L. Vázquez, Fast diffusion flow on manifolds of nonpositive curvature, J. Evol. Equ., 8 (2008), 99-128.doi: 10.1007/s00028-007-0345-4. |
[12] |
M. Bonforte, G. Grillo and J. L. Vázquez, Special fast diffusion with slow asymptotics: Entropy method and flow on a Riemann manifold, Arch. Rat. Mech. Anal., 196 (2010), 631-680.doi: 10.1007/s00205-009-0252-7. |
[13] |
S. M. Buckley and P. Koskela, New Poincaré inequalities from old, Ann. Acad. Sci. Fenn. Math., 23 (1998), 251-260. |
[14] |
S.-K. Chua and R. L. Wheeden, Sharp conditions for weighted 1-dimensional Poincaré inequalities, Indiana Univ. Math. J., 49 (2000), 143-175.doi: 10.1512/iumj.2000.49.1754. |
[15] |
S.-K. Chua and R. L. Wheeden, Weighted Poincaré inequalities on convex domains, Math. Res. Lett., 17 (2010), 993-1011. |
[16] |
E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989.doi: 10.1017/CBO9780511566158. |
[17] |
E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22.doi: 10.1515/crll.1985.357.1. |
[18] |
J. Dolbeault, I. Gentil, A. Guillin and F.-Y. Wang, $L^q$-functional inequalities and weighted porous media equations, Potential Anal., 28 (2008), 35-59.doi: 10.1007/s11118-007-9066-0. |
[19] |
J. Dolbeault, B. Nazaret and G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations, Commun. Math. Sci., 6 (2008), 477-494. |
[20] |
D. E. Edmunds and B. Opic, Weighted Poincaré and Friedrichs inequalities, J. London Math. Soc. (2), 47 (1993), 79-96.doi: 10.1112/jlms/s2-47.1.79. |
[21] |
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. |
[22] |
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: 10.2307/2160476. |
[23] |
E. Fabes, M. Fukushima, L. Gross, C. Kenig, M. Röckner and D. W. Stroock, "Dirichlet Forms," Lectures given at the First C.I.M.E. Session held in Varenna, June 8-19, 1992, Edited by G. Dell'Antonio and U. Mosco, Lecture Notes in Mathematics, 1563, Springer-Verlag, Berlin, 1993. |
[24] |
P. Federbush, A partial alternate derivation of a result of Nelson, J. Math. Phys., 10 (1969), 50-52. |
[25] |
S. Filippas, L. Moschini and A. Tertikas, Sharp two-sided heat kernel estimates for critical Schrödinger operators on bounded domains, Comm. Math. Phys., 273 (2007), 237-281.doi: 10.1007/s00220-007-0253-z. |
[26] |
M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions," Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser Boston, Inc., Boston, MA, 2010.doi: 10.1007/978-0-8176-4651-6. |
[27] |
A. Grigor'yan, Heat kernels on weighted manifolds and applications, in "The Ubiquitous Heat Kernel," Contemp. Math., 398, Amer. Math. Soc., Providence, RI, (2006), 93-191.doi: 10.1090/conm/398/07486. |
[28] |
A. Grigor'yan, "Heat Kernel and Analysis on Manifolds," AMS/IP Studies in Advanced Mathematics, 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. |
[29] |
G. Grillo, On the equivalence between $p$-Poincaré inequalities and $L^r$-$L^q$ regularization and decay estimates of certain nonlinear evolutions, J. Differential Equations, 249 (2010), 2561-2576.doi: 10.1016/j.jde.2010.05.022. |
[30] |
L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083. |
[31] |
E. Hebey, "Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities," Courant Lecture Notes in Mathematics, 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. |
[32] |
R. Hurri, The weighted Poincaré inequalities, Math. Scand., 67 (1990), 145-160. |
[33] |
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. |
[34] |
S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 34 (1981), 831-852.doi: 10.1002/cpa.3160340605. |
[35] |
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. |
[36] |
A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin., 25 (1984), 537-554. |
[37] |
A. Kufner and B. Opic, "Hardy-Type Inequalities," Pitman Research Notes in Mathematics Series, 219, Longman Scientific & Technical, Harlow, 1990. |
[38] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968. |
[39] |
G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. |
[40] |
V. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. |
[41] |
B. Muckenhoupt, Hardy's inequality with weights, Studia Math., 44 (1972), 31-38. |
[42] |
B. Muckenhoupt, Weighted normed inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. |
[43] |
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. |
[44] |
O. A. Oleĭnik, On the equations of unsteady filtration, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 1210-1213. |
[45] |
O. A. Oleĭnik, A. S. Kalašnikov and Y.-L. Čžou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat., 22 (1958), 667-704. |
[46] |
M. M. Porzio, On decay estimates, J. Evol. Equ., 9 (2009), 561-591.doi: 10.1007/s00028-009-0024-8. |
[47] |
M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.doi: 10.1006/jdeq.1993.1045. |
[48] |
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. |
[49] |
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. |
[50] |
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. |
[51] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. |
[52] |
J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type," Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006.doi: 10.1093/acprof:oso/9780199202973.001.0001. |
[53] |
J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[54] |
F.-Y. Wang, Orlicz-Poincaré inequalities, Proc. Edinb. Math. Soc. (2), 51 (2008), 529-543.doi: 10.1017/S0013091506000526. |