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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  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 & Control Theory, 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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