August  2013, 33(8): 3599-3640. doi: 10.3934/dcds.2013.33.3599

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

Received  August 2012 Revised  November 2012 Published  January 2013

We study weighted porous media equations on domains $\Omega\subseteq{\mathbb R}^N$, either with Dirichlet or with Neumann homogeneous boundary conditions when $\Omega\not={\mathbb R}^N$. Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, $L^{q_0}$-$L^\varrho$ smoothing effects ($1\leq q_0<\varrho<\infty$) are discussed for short time, in connection with the validity of a Poincaré inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. In fact, we prove full equivalence between certain $L^{q_0}$-$L^\varrho$ smoothing effects and suitable weighted Poincaré-type inequalities. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case $\Omega={\mathbb R}^N$ when the corresponding weight makes its measure finite, so that solutions converge to their weighted mean value instead than to zero. Examples are given in terms of wide classes of weights.
Citation: Gabriele Grillo, Matteo Muratori, Maria Michaela Porzio. Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3599-3640. doi: 10.3934/dcds.2013.33.3599
References:
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R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. M. Buckley and P. Koskela, New Poincaré inequalities from old,, Ann. Acad. Sci. Fenn. Math., 23 (1998), 251.   Google Scholar

[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.  Google Scholar

[15]

S.-K. Chua and R. L. Wheeden, Weighted Poincaré inequalities on convex domains,, Math. Res. Lett., 17 (2010), 993.   Google Scholar

[16]

E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Tracts in Mathematics, 92 (1989).  doi: 10.1017/CBO9780511566158.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Grigor'yan, Heat kernels on weighted manifolds and applications,, in, 398 (2006), 93.  doi: 10.1090/conm/398/07486.  Google Scholar

[28]

A. Grigor'yan, "Heat Kernel and Analysis on Manifolds,", AMS/IP Studies in Advanced Mathematics, 47 (2009).   Google Scholar

[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.  Google Scholar

[30]

L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.   Google Scholar

[31]

E. Hebey, "Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities,", Courant Lecture Notes in Mathematics, 5 (1999).   Google Scholar

[32]

R. Hurri, The weighted Poincaré inequalities,, Math. Scand., 67 (1990), 145.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces,, Comment. Math. Univ. Carolin., 25 (1984), 537.   Google Scholar

[37]

A. Kufner and B. Opic, "Hardy-Type Inequalities,", Pitman Research Notes in Mathematics Series, 219 (1990).   Google Scholar

[38]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968).   Google Scholar

[39]

G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996).   Google Scholar

[40]

V. Maz'ja, "Sobolev Spaces,", Springer Series in Soviet Mathematics, (1985).   Google Scholar

[41]

B. Muckenhoupt, Hardy's inequality with weights,, Studia Math., 44 (1972), 31.   Google Scholar

[42]

B. Muckenhoupt, Weighted normed inequalities for the Hardy maximal function,, Trans. Amer. Math. Soc., 165 (1972), 207.   Google Scholar

[43]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115.   Google Scholar

[44]

O. A. Oleĭnik, On the equations of unsteady filtration,, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 1210.   Google Scholar

[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.   Google Scholar

[46]

M. M. Porzio, On decay estimates,, J. Evol. Equ., 9 (2009), 561.  doi: 10.1007/s00028-009-0024-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).   Google Scholar

[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.  Google Scholar

[53]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (2007).   Google Scholar

[54]

F.-Y. Wang, Orlicz-Poincaré inequalities,, Proc. Edinb. Math. Soc. (2), 51 (2008), 529.  doi: 10.1017/S0013091506000526.  Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. M. Buckley and P. Koskela, New Poincaré inequalities from old,, Ann. Acad. Sci. Fenn. Math., 23 (1998), 251.   Google Scholar

[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.  Google Scholar

[15]

S.-K. Chua and R. L. Wheeden, Weighted Poincaré inequalities on convex domains,, Math. Res. Lett., 17 (2010), 993.   Google Scholar

[16]

E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge Tracts in Mathematics, 92 (1989).  doi: 10.1017/CBO9780511566158.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. Grigor'yan, Heat kernels on weighted manifolds and applications,, in, 398 (2006), 93.  doi: 10.1090/conm/398/07486.  Google Scholar

[28]

A. Grigor'yan, "Heat Kernel and Analysis on Manifolds,", AMS/IP Studies in Advanced Mathematics, 47 (2009).   Google Scholar

[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.  Google Scholar

[30]

L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.   Google Scholar

[31]

E. Hebey, "Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities,", Courant Lecture Notes in Mathematics, 5 (1999).   Google Scholar

[32]

R. Hurri, The weighted Poincaré inequalities,, Math. Scand., 67 (1990), 145.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces,, Comment. Math. Univ. Carolin., 25 (1984), 537.   Google Scholar

[37]

A. Kufner and B. Opic, "Hardy-Type Inequalities,", Pitman Research Notes in Mathematics Series, 219 (1990).   Google Scholar

[38]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1968).   Google Scholar

[39]

G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996).   Google Scholar

[40]

V. Maz'ja, "Sobolev Spaces,", Springer Series in Soviet Mathematics, (1985).   Google Scholar

[41]

B. Muckenhoupt, Hardy's inequality with weights,, Studia Math., 44 (1972), 31.   Google Scholar

[42]

B. Muckenhoupt, Weighted normed inequalities for the Hardy maximal function,, Trans. Amer. Math. Soc., 165 (1972), 207.   Google Scholar

[43]

L. Nirenberg, On elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115.   Google Scholar

[44]

O. A. Oleĭnik, On the equations of unsteady filtration,, Dokl. Akad. Nauk SSSR (N.S.), 113 (1957), 1210.   Google Scholar

[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.   Google Scholar

[46]

M. M. Porzio, On decay estimates,, J. Evol. Equ., 9 (2009), 561.  doi: 10.1007/s00028-009-0024-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).   Google Scholar

[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.  Google Scholar

[53]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,", Oxford Mathematical Monographs, (2007).   Google Scholar

[54]

F.-Y. Wang, Orlicz-Poincaré inequalities,, Proc. Edinb. Math. Soc. (2), 51 (2008), 529.  doi: 10.1017/S0013091506000526.  Google Scholar

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