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On the moments of solutions to linear parabolic equations involving the biharmonic operator
Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities
1. | Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy, Italy |
2. | Dipartimento di Matematica, Università di Roma "La Sapienza", Piazzale A. Moro 2, 00185 Roma, Italy |
References:
[1] |
R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (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.
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.
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.
|
[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.
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.
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.
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.
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.
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, 61 (2005), 15.
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.
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.
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.
|
[14] |
S.-K. Chua and R. L. Wheeden, Sharp conditions for weighted 1-dimensional Poincaré inequalities,, Indiana Univ. Math. J., 49 (2000), 143.
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.
|
[16] |
E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Tracts in Mathematics, 92 (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.
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.
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.
|
[20] |
D. E. Edmunds and B. Opic, Weighted Poincaré and Friedrichs inequalities,, J. London Math. Soc. (2), 47 (1993), 79.
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.
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.
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 (1993).
|
[24] |
P. Federbush, A partial alternate derivation of a result of Nelson,, J. Math. Phys., 10 (1969), 50. Google Scholar |
[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.
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 (2010).
doi: 10.1007/978-0-8176-4651-6. |
[27] |
A. Grigor'yan, Heat kernels on weighted manifolds and applications,, in, 398 (2006), 93.
doi: 10.1090/conm/398/07486. |
[28] |
A. Grigor'yan, "Heat Kernel and Analysis on Manifolds,", AMS/IP Studies in Advanced Mathematics, 47 (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.
doi: 10.1016/j.jde.2010.05.022. |
[30] |
L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.
|
[31] |
E. Hebey, "Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities,", Courant Lecture Notes in Mathematics, 5 (1999).
|
[32] |
R. Hurri, The weighted Poincaré inequalities,, Math. Scand., 67 (1990), 145.
|
[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.
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.
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.
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.
|
[37] |
A. Kufner and B. Opic, "Hardy-Type Inequalities,", Pitman Research Notes in Mathematics Series, 219 (1990).
|
[38] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968).
|
[39] |
G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996).
|
[40] |
V. Maz'ja, "Sobolev Spaces,", Springer Series in Soviet Mathematics, (1985).
|
[41] |
B. Muckenhoupt, Hardy's inequality with weights,, Studia Math., 44 (1972), 31.
|
[42] |
B. Muckenhoupt, Weighted normed inequalities for the Hardy maximal function,, Trans. Amer. Math. Soc., 165 (1972), 207.
|
[43] |
L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115.
|
[44] |
O. A. Oleĭnik, On the equations of unsteady filtration,, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 1210.
|
[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.
|
[46] |
M. M. Porzio, On decay estimates,, J. Evol. Equ., 9 (2009), 561.
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.
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.
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.
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.
doi: 10.3934/cpaa.2009.8.493. |
[51] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (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 (2006).
doi: 10.1093/acprof:oso/9780199202973.001.0001. |
[53] |
J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (2007).
|
[54] |
F.-Y. Wang, Orlicz-Poincaré inequalities,, Proc. Edinb. Math. Soc. (2), 51 (2008), 529.
doi: 10.1017/S0013091506000526. |
show all references
References:
[1] |
R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (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.
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.
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.
|
[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.
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.
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.
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.
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.
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, 61 (2005), 15.
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.
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.
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.
|
[14] |
S.-K. Chua and R. L. Wheeden, Sharp conditions for weighted 1-dimensional Poincaré inequalities,, Indiana Univ. Math. J., 49 (2000), 143.
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.
|
[16] |
E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Tracts in Mathematics, 92 (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.
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.
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.
|
[20] |
D. E. Edmunds and B. Opic, Weighted Poincaré and Friedrichs inequalities,, J. London Math. Soc. (2), 47 (1993), 79.
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.
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.
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 (1993).
|
[24] |
P. Federbush, A partial alternate derivation of a result of Nelson,, J. Math. Phys., 10 (1969), 50. Google Scholar |
[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.
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 (2010).
doi: 10.1007/978-0-8176-4651-6. |
[27] |
A. Grigor'yan, Heat kernels on weighted manifolds and applications,, in, 398 (2006), 93.
doi: 10.1090/conm/398/07486. |
[28] |
A. Grigor'yan, "Heat Kernel and Analysis on Manifolds,", AMS/IP Studies in Advanced Mathematics, 47 (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.
doi: 10.1016/j.jde.2010.05.022. |
[30] |
L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.
|
[31] |
E. Hebey, "Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities,", Courant Lecture Notes in Mathematics, 5 (1999).
|
[32] |
R. Hurri, The weighted Poincaré inequalities,, Math. Scand., 67 (1990), 145.
|
[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.
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.
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.
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.
|
[37] |
A. Kufner and B. Opic, "Hardy-Type Inequalities,", Pitman Research Notes in Mathematics Series, 219 (1990).
|
[38] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968).
|
[39] |
G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996).
|
[40] |
V. Maz'ja, "Sobolev Spaces,", Springer Series in Soviet Mathematics, (1985).
|
[41] |
B. Muckenhoupt, Hardy's inequality with weights,, Studia Math., 44 (1972), 31.
|
[42] |
B. Muckenhoupt, Weighted normed inequalities for the Hardy maximal function,, Trans. Amer. Math. Soc., 165 (1972), 207.
|
[43] |
L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115.
|
[44] |
O. A. Oleĭnik, On the equations of unsteady filtration,, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 1210.
|
[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.
|
[46] |
M. M. Porzio, On decay estimates,, J. Evol. Equ., 9 (2009), 561.
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.
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.
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.
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.
doi: 10.3934/cpaa.2009.8.493. |
[51] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (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 (2006).
doi: 10.1093/acprof:oso/9780199202973.001.0001. |
[53] |
J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (2007).
|
[54] |
F.-Y. Wang, Orlicz-Poincaré inequalities,, Proc. Edinb. Math. Soc. (2), 51 (2008), 529.
doi: 10.1017/S0013091506000526. |
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