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June  2022, 11(3): 793-825. doi: 10.3934/eect.2021026

Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation

1. 

Department of Mathematics, University of Patras, 26504 Rio Patras, Greece

2. 

Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA

* Corresponding author: Yuanzhen Shao

Received  September 2020 Revised  April 2021 Published  June 2022 Early access  May 2021

The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincaré, Nash, Super Poincaré and logarithmic Sobolev type inequalities, on complete Riemannian manifolds satisfying some mild geometric assumptions. Second, based on the derived nonlocal functional inequalities, we analyze the asymptotic behavior of the solution to the fractional porous medium equation, $ \partial_t u +(-\Delta)^\sigma (|u|^{m-1}u ) = 0 $ with $ m>0 $ and $ \sigma\in (0, 1) $. In addition, we establish the global well-posedness of the equation on an arbitrary complete Riemannian manifold.

Citation: Nikolaos Roidos, Yuanzhen Shao. Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation. Evolution Equations and Control Theory, 2022, 11 (3) : 793-825. doi: 10.3934/eect.2021026
References:
[1]

N. D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. J. Math., 30 (1981), 749-785.  doi: 10.1512/iumj.1981.30.30056.

[2]

A. Alphonse and C. M. Elliott, Well-posedness of a fractional porous medium equation on an evolving surface, Nonlinear Anal., 137 (2016), 3-42.  doi: 10.1016/j.na.2016.01.010.

[3]

H. AmannM. Hieber and G. Simonett, Bounded $H_\infty$-calculus for elliptic operators, Differential Integral Equations, 7 (1994), 613-653. 

[4]

H. Amann, Linear and Quasilinear Parabolic Problems: Volume I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[5]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184. Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[6]

V. BanicaM. d. M. González and M. Sáez, Some constructions for the fractional Laplacian on noncompact manifolds, Rev. Mat. Iberoam, 31 (2015), 681-712.  doi: 10.4171/RMI/850.

[7]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Translated from the Romanian. Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976.

[8]

P. Bénilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations, In Contributions to Analysis and Geometry (Baltimore, Md., 1980), pp. 23{39, Johns Hopkins Univ. Press, Baltimore, Md., 1981.

[9]

P. Bénilan and R. Gariepy, Strong solutions in $L^1$ of degenerate parabolic equations, J. Differential Equations, 119 (1995), 473-502.  doi: 10.1006/jdeq.1995.1099.

[10]

E. Berchio, M. Bonforte, D. Ganguly and G. Grillo, The fractional porous medium equation on the hyperbolic space, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 169. doi: 10.1007/s00526-020-01817-2.

[11]

M. BonforteA. Figalli and X. Ros-Oton, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains, Comm. Pure Appl. Math., 70 (2017), 1472-1508.  doi: 10.1002/cpa.21673.

[12]

M. BonforteA. Figalli and J. L. Vázquez, Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains, Anal. PDE, 11 (2018), 945-982.  doi: 10.2140/apde.2018.11.945.

[13]

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.

[14]

M. BonforteG. 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.

[15]

M. BonforteY. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.

[16]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Arch. Ration. Mech. Anal., 218 (2015), 317-362.  doi: 10.1007/s00205-015-0861-2.

[17]

M. Bonforte and J. L. Vázquez, Fractional nonlinear degenerate diffusion equations on bounded domains part I. Existence, uniqueness and upper bounds, Nonlinear Anal., 131 (2016), 363-398.  doi: 10.1016/j.na.2015.10.005.

[18]

H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.

[19]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[20]

E. A. Carlen and M. Loss, Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier-Stokes equation, Duke Math. J., 81 (1995), 135-157.  doi: 10.1215/S0012-7094-95-08110-1.

[21]

F. Cipriania and G. Grillo, Uniform bounds for solutions to quasilinear parabolic equations, J. Diff. Equations, 177 (2001), 209-234.  doi: 10.1006/jdeq.2000.3985.

[22]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298.  doi: 10.2307/2373376.

[23]

C. M. Dafermos, Asymptotic behavior of solutions of evolution equations, in Nonlinear Evolution Equations, (M. G. Crandall, Ed.) Academic Press, New York, 1978.

[24]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.

[25]

A. Grigor'yan, Heat kernel upper bounds on a complete non-compact manifold, Revista Mathemática Iberoamericana, 10 (1994), 395-452.  doi: 10.4171/RMI/157.

[26]

A. Grigor'yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom., 45 (1997), 33-52.  doi: 10.4310/jdg/1214459753.

[27]

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. doi: 10.1090/amsip/047.

[28]

G. GrilloM. Muratori and F. Punzo, On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density, Discrete Contin. Dyn. Syst., 35 (2015), 5927-5962.  doi: 10.3934/dcds.2015.35.5927.

[29]

G. GrilloK. Ishige and M. Muratori, Nonlinear characterizations of stochastic completeness, J. Math. Pures Appl., 139 (2020), 63-82.  doi: 10.1016/j.matpur.2020.05.008.

[30]

G. Grillo and M. Muratori, Radial fast diffusion on the hyperbolic space, Proc. London Math. Soc., 109 (2014), 283-317.  doi: 10.1112/plms/pdt071.

[31]

G. Grillo and M. Muratori, Smoothing effects for the porous medium equation on Cartan-Hadamard manifolds, Nonlinear Anal., 131 (2016), 346-362.  doi: 10.1016/j.na.2015.07.029.

[32]

G. GrilloM. Muratori and F. Punzo, The porous medium equation with measure data on negatively curved Riemannian manifolds, J. Eur. Math. Soc., 20 (2018), 2769-2812.  doi: 10.4171/JEMS/824.

[33]

G. GrilloM. Muratori and F. Punzo, The porous medium equation with large initial data on negatively curved Riemannian manifolds, J. Math. Pures Appl., 113 (2018), 195-226.  doi: 10.1016/j.matpur.2017.07.021.

[34]

G. GrilloM. Muratori and J. L. Vázquez, The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour, Adv. Math., 314 (2017), 328-377.  doi: 10.1016/j.aim.2017.04.023.

[35]

G. GrilloM. Muratori and J. L. Vázquez, The porous medium equation on Riemannian manifolds with negative curvature: The superquadratic case, Math. Ann., 373 (2019), 119-153.  doi: 10.1007/s00208-018-1680-1.

[36]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math, 97 (1975), 1061-1083.  doi: 10.2307/2373688.

[37]

V. A. Liskevich and Y. A. Semenov, Some problems on Markov semigroups, In Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Math. Top., 11, Adv. Partial Differential Equations, Akademie Verlag, Berlin, 1996,163–217.

[38]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017.

[39]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Applied Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408.

[40]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44. Springer-Verlag, New York. 1983. doi: 10.1007/978-1-4612-5561-1.

[41]

F. Punzo and G. Terrone, On the Cauchy problem for a general fractional porous medium equation with variable density, Nonlinear Anal., 98 (2014), 27-47.  doi: 10.1016/j.na.2013.12.007.

[42]

M. Röckner and F.-Y. Wang, Weak Poincaré inequalities and $L^2$-convergence rates of Markov semigroups, J. Funct. Anal., 185 (2001), 564-603.  doi: 10.1006/jfan.2001.3776.

[43]

N. Roidos and Y. Shao, The fractional porous medium equation on manifolds with conical singularities, arXiv: 1908.06915.

[44]

N. Roidos and Y. Shao, The fractional porous medium equation on manifolds with conical singularities II, arXiv: 1908.07138v3.

[45]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, 2$^{nd}$ edition. De Gruyter Studies in Mathematics, 37. Walter de Gruyter & Co., Berlin, 2012. doi: 10.1515/9783110269338.

[46]

R. L. Schilling and J. Wang, Functional inequalities and subordination: Stability of Nash and Poincaré inequalities, Math. Z., 272 (2012), 921-936.  doi: 10.1007/s00209-011-0964-x.

[47]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[48]

H. Tanabe, Equations of Evolution, Monographs and studies in mathematics 6, Pitman Publishing, 1979.

[49] N. T. VaropoulosL. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, 1992. 
[50]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

[51]

J. L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type, J. Eur. Math. Soc. (JEMS), 16 (2014), 769-803.  doi: 10.4171/JEMS/446.

[52]

J. L. Vázquez, The mesa problem for the fractional porous medium equation, Interfaces Free Bound., 17 (2015), 261-286.  doi: 10.4171/IFB/342.

[53]

J. L. Vázquez, Fundamental solution and long time behavior of the porous medium equation in hyperbolic space, J. Math. Pures Appl., 104 (2015), 454-484.  doi: 10.1016/j.matpur.2015.03.005.

show all references

References:
[1]

N. D. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. J. Math., 30 (1981), 749-785.  doi: 10.1512/iumj.1981.30.30056.

[2]

A. Alphonse and C. M. Elliott, Well-posedness of a fractional porous medium equation on an evolving surface, Nonlinear Anal., 137 (2016), 3-42.  doi: 10.1016/j.na.2016.01.010.

[3]

H. AmannM. Hieber and G. Simonett, Bounded $H_\infty$-calculus for elliptic operators, Differential Integral Equations, 7 (1994), 613-653. 

[4]

H. Amann, Linear and Quasilinear Parabolic Problems: Volume I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[5]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184. Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[6]

V. BanicaM. d. M. González and M. Sáez, Some constructions for the fractional Laplacian on noncompact manifolds, Rev. Mat. Iberoam, 31 (2015), 681-712.  doi: 10.4171/RMI/850.

[7]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Translated from the Romanian. Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976.

[8]

P. Bénilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations, In Contributions to Analysis and Geometry (Baltimore, Md., 1980), pp. 23{39, Johns Hopkins Univ. Press, Baltimore, Md., 1981.

[9]

P. Bénilan and R. Gariepy, Strong solutions in $L^1$ of degenerate parabolic equations, J. Differential Equations, 119 (1995), 473-502.  doi: 10.1006/jdeq.1995.1099.

[10]

E. Berchio, M. Bonforte, D. Ganguly and G. Grillo, The fractional porous medium equation on the hyperbolic space, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 169. doi: 10.1007/s00526-020-01817-2.

[11]

M. BonforteA. Figalli and X. Ros-Oton, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains, Comm. Pure Appl. Math., 70 (2017), 1472-1508.  doi: 10.1002/cpa.21673.

[12]

M. BonforteA. Figalli and J. L. Vázquez, Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains, Anal. PDE, 11 (2018), 945-982.  doi: 10.2140/apde.2018.11.945.

[13]

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.

[14]

M. BonforteG. 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.

[15]

M. BonforteY. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.

[16]

M. Bonforte and J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Arch. Ration. Mech. Anal., 218 (2015), 317-362.  doi: 10.1007/s00205-015-0861-2.

[17]

M. Bonforte and J. L. Vázquez, Fractional nonlinear degenerate diffusion equations on bounded domains part I. Existence, uniqueness and upper bounds, Nonlinear Anal., 131 (2016), 363-398.  doi: 10.1016/j.na.2015.10.005.

[18]

H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.

[19]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[20]

E. A. Carlen and M. Loss, Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier-Stokes equation, Duke Math. J., 81 (1995), 135-157.  doi: 10.1215/S0012-7094-95-08110-1.

[21]

F. Cipriania and G. Grillo, Uniform bounds for solutions to quasilinear parabolic equations, J. Diff. Equations, 177 (2001), 209-234.  doi: 10.1006/jdeq.2000.3985.

[22]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298.  doi: 10.2307/2373376.

[23]

C. M. Dafermos, Asymptotic behavior of solutions of evolution equations, in Nonlinear Evolution Equations, (M. G. Crandall, Ed.) Academic Press, New York, 1978.

[24]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.

[25]

A. Grigor'yan, Heat kernel upper bounds on a complete non-compact manifold, Revista Mathemática Iberoamericana, 10 (1994), 395-452.  doi: 10.4171/RMI/157.

[26]

A. Grigor'yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom., 45 (1997), 33-52.  doi: 10.4310/jdg/1214459753.

[27]

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. doi: 10.1090/amsip/047.

[28]

G. GrilloM. Muratori and F. Punzo, On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density, Discrete Contin. Dyn. Syst., 35 (2015), 5927-5962.  doi: 10.3934/dcds.2015.35.5927.

[29]

G. GrilloK. Ishige and M. Muratori, Nonlinear characterizations of stochastic completeness, J. Math. Pures Appl., 139 (2020), 63-82.  doi: 10.1016/j.matpur.2020.05.008.

[30]

G. Grillo and M. Muratori, Radial fast diffusion on the hyperbolic space, Proc. London Math. Soc., 109 (2014), 283-317.  doi: 10.1112/plms/pdt071.

[31]

G. Grillo and M. Muratori, Smoothing effects for the porous medium equation on Cartan-Hadamard manifolds, Nonlinear Anal., 131 (2016), 346-362.  doi: 10.1016/j.na.2015.07.029.

[32]

G. GrilloM. Muratori and F. Punzo, The porous medium equation with measure data on negatively curved Riemannian manifolds, J. Eur. Math. Soc., 20 (2018), 2769-2812.  doi: 10.4171/JEMS/824.

[33]

G. GrilloM. Muratori and F. Punzo, The porous medium equation with large initial data on negatively curved Riemannian manifolds, J. Math. Pures Appl., 113 (2018), 195-226.  doi: 10.1016/j.matpur.2017.07.021.

[34]

G. GrilloM. Muratori and J. L. Vázquez, The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour, Adv. Math., 314 (2017), 328-377.  doi: 10.1016/j.aim.2017.04.023.

[35]

G. GrilloM. Muratori and J. L. Vázquez, The porous medium equation on Riemannian manifolds with negative curvature: The superquadratic case, Math. Ann., 373 (2019), 119-153.  doi: 10.1007/s00208-018-1680-1.

[36]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math, 97 (1975), 1061-1083.  doi: 10.2307/2373688.

[37]

V. A. Liskevich and Y. A. Semenov, Some problems on Markov semigroups, In Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Math. Top., 11, Adv. Partial Differential Equations, Akademie Verlag, Berlin, 1996,163–217.

[38]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.  doi: 10.1016/j.aim.2010.07.017.

[39]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Applied Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408.

[40]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44. Springer-Verlag, New York. 1983. doi: 10.1007/978-1-4612-5561-1.

[41]

F. Punzo and G. Terrone, On the Cauchy problem for a general fractional porous medium equation with variable density, Nonlinear Anal., 98 (2014), 27-47.  doi: 10.1016/j.na.2013.12.007.

[42]

M. Röckner and F.-Y. Wang, Weak Poincaré inequalities and $L^2$-convergence rates of Markov semigroups, J. Funct. Anal., 185 (2001), 564-603.  doi: 10.1006/jfan.2001.3776.

[43]

N. Roidos and Y. Shao, The fractional porous medium equation on manifolds with conical singularities, arXiv: 1908.06915.

[44]

N. Roidos and Y. Shao, The fractional porous medium equation on manifolds with conical singularities II, arXiv: 1908.07138v3.

[45]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, 2$^{nd}$ edition. De Gruyter Studies in Mathematics, 37. Walter de Gruyter & Co., Berlin, 2012. doi: 10.1515/9783110269338.

[46]

R. L. Schilling and J. Wang, Functional inequalities and subordination: Stability of Nash and Poincaré inequalities, Math. Z., 272 (2012), 921-936.  doi: 10.1007/s00209-011-0964-x.

[47]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[48]

H. Tanabe, Equations of Evolution, Monographs and studies in mathematics 6, Pitman Publishing, 1979.

[49] N. T. VaropoulosL. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, 1992. 
[50]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

[51]

J. L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type, J. Eur. Math. Soc. (JEMS), 16 (2014), 769-803.  doi: 10.4171/JEMS/446.

[52]

J. L. Vázquez, The mesa problem for the fractional porous medium equation, Interfaces Free Bound., 17 (2015), 261-286.  doi: 10.4171/IFB/342.

[53]

J. L. Vázquez, Fundamental solution and long time behavior of the porous medium equation in hyperbolic space, J. Math. Pures Appl., 104 (2015), 454-484.  doi: 10.1016/j.matpur.2015.03.005.

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