The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

Agueh, 2003, Asymptotic behavior for doubly degenerate parabolic equations, C. R. Math. Acad. Sci. Paris, 337, 331, 10.1016/S1631-073X(03)00352-2 Agueh, 2005, Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory, Adv. Differ. Equ., 10, 309 Agueh, 2008, Rates of decay to equilibria for -Laplacian type equations, Nonlinear Anal., Theory Methods Appl., 68, 1909, 10.1016/j.na.2007.01.043 Agueh, 2009, Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime, J. Evol. Equ., 10, 59, 10.1007/s00028-009-0040-8 Ambrosio, 2021, 10.1007/978-3-030-72162-6 Ambrosio, 2008 Andreu-Vaillo, 2004 Aronson, 1968, Non-negative solutions of linear parabolic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3), 22, 607 Audrito, 2017, The Fisher-KPP problem with doubly nonlinear “fast” diffusion, Nonlinear Anal., Theory Methods Appl., 157, 212, 10.1016/j.na.2017.03.015 Barenblatt, 2004, Nonlinear diffusion and image contour enhancement, Interfaces Free Bound., 6, 31, 10.4171/IFB/90 P. Bénilan, M.G. Crandall, A. Pazy, Nonlinear Evolution Equations in Banach Spaces, Unpublished book. Bénilan, 1981, Regularizing effects of homogeneous evolution equations, 23 Bernis, 1988, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279, 373, 10.1007/BF01456275 Blanchet, 2009, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., 191, 347, 10.1007/s00205-008-0155-z Bögelein, 2020, Higher integrability for doubly nonlinear parabolic systems, J. Math. Pures Appl. Neuvième Série, 143, 31, 10.1016/j.matpur.2020.06.009 Bögelein, 2018, Doubly nonlinear equations of porous medium type, Arch. Ration. Mech. Anal., 229, 503, 10.1007/s00205-018-1221-9 Bonforte, 2017, Weighted fast diffusion equations (part I): sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities, Kinet. Relat. Models, 10, 33, 10.3934/krm.2017002 Bonforte, 2017, Weighted fast diffusion equations (part II): sharp asymptotic rates of convergence in relative error by entropy methods, Kinet. Relat. Models, 10, 61, 10.3934/krm.2017003 Bonforte Bonforte Bonforte, 2012, Total variation flow and sign fast diffusion in one dimension, J. Differ. Equ., 252, 4455, 10.1016/j.jde.2012.01.003 Bonforte, 2006, Super and ultracontractive bounds for doubly nonlinear evolution equations, Rev. Mat. Iberoam., 22, 111 Bonforte, 2010, Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation, Adv. Math., 224, 2151, 10.1016/j.aim.2010.01.023 Bonforte, 2019, Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity, Adv. Math., 345, 1075, 10.1016/j.aim.2019.01.018 Bonforte Bonforte, 2017, Optimal existence and uniqueness theory for the fractional heat equation, Nonlinear Anal., Theory Methods Appl., 153, 142, 10.1016/j.na.2016.08.027 Bonforte, 2006, Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., 240, 399, 10.1016/j.jfa.2006.07.009 Brézis, 1973 Brézis, 1983, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. Neuvième Série, 62, 73 Recalde, 2015, Harnack estimates at large: sharp pointwise estimates for nonnegative solutions to a class of singular parabolic equations, Nonlinear Anal., Theory Methods Appl., 121, 153, 10.1016/j.na.2015.03.003 Carrillo, 2022, A fast regularisation of a Newtonian vortex equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 39, 705, 10.4171/aihpc/17 Carrillo, 2003, Fine asymptotics for fast diffusion equations, Commun. Partial Differ. Equ., 28, 1023, 10.1081/PDE-120021185 Cipriani, 2001, Uniform bounds for solutions to quasilinear parabolic equations, J. Differ. Equ., 177, 209, 10.1006/jdeq.2000.3985 Cipriani, 2002, Lq−L∞ Hölder continuity for quasilinear parabolic equations associated to Sobolev derivations, J. Math. Anal. Appl., 270, 267, 10.1016/S0022-247X(02)00084-7 Crandall, 1971, Generation of semi-groups of nonlinear transformations on general Banach spaces, Am. J. Math., 93, 265, 10.2307/2373376 Del Pino, 2002, Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the p-Laplacian, C. R. Math. Acad. Sci. Paris, 334, 365, 10.1016/S1631-073X(02)02225-2 DiBenedetto, 1993 DiBenedetto, 1984, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math., 1984, 83, 10.1515/crll.1984.349.83 DiBenedetto, 1985, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357, 1 DiBenedetto, 2008, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math., 200, 181, 10.1007/s11511-008-0026-3 DiBenedetto, 2012 DiBenedetto, 2007, Intrinsic Harnack inequalities for quasi-linear singular parabolic partial differential equation, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. Serie IX. Rend. Lincei, Mat. Appl., 18, 359, 10.4171/RLM/502 DiBenedetto, 1989, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Am. Math. Soc., 314, 187, 10.1090/S0002-9947-1989-0962278-5 DiBenedetto, 1990, Non-negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when 1<p<2, Arch. Ration. Mech. Anal., 111, 225, 10.1007/BF00400111 Duzaar, 2010, Gradient continuity estimates, Calc. Var. Partial Differ. Equ., 39, 379, 10.1007/s00526-010-0314-6 Duzaar, 2010, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., 259, 2961, 10.1016/j.jfa.2010.08.006 Ebmeyer, 2005, The smoothing property for a class of doubly nonlinear parabolic equations, Trans. Am. Math. Soc., 357, 3239, 10.1090/S0002-9947-05-03790-6 Esteban, 1990, Régularité des solutions positives de l'équation parabolique p-Laplacienne. (Regularity of nonnegative solutions of the p-Laplacian parabolic equation), C. R. Acad. Sci. Paris, Ser. I, 310, 105 Fino, 2014, Conservation of the mass for solutions to a class of singular parabolic equations, Kodai Math. J., 37, 519, 10.2996/kmj/1414674606 Fornaro, 2013, Energy estimates and integral Harnack inequality for some doubly nonlinear singular parabolic equations, 179 Fornaro, 2014, Llocr−Lloc∞ estimates and expansion of positivity for a class of doubly non linear singular parabolic equations, Discrete Contin. Dyn. Syst., Ser. S, 7, 737 Friedman, 1980, The asymptotic behaviour of gas in an n-dimensional porous medium, Trans. Am. Math. Soc., 262, 551 Gagliardo, 1958, Proprietà di alcune classi di funzioni in più variabili, Ric. Mat., 7, 102 Galaktionov, 2004, vol. 56 Hadamard, 1955, Extension à l'équation de la chaleur d'un théorème de A. Harnack, Rend. Circ. Mat. Palermo, Serie II, 3, 337 Herrero, 1982, On the propagation properties of a nonlinear degenerate parabolic equation, Commun. Partial Differ. Equ., 7, 1381, 10.1080/03605308208820255 Hynd, 2016, A doubly nonlinear evolution for the optimal Poincaré inequality, Calc. Var. Partial Differ. Equ., 55, 10.1007/s00526-016-1026-3 Hynd, 2016, Hölder estimates and large time behaviour for a nonlocal doubly nonlinear evolution, Anal. PDE, 9, 1447, 10.2140/apde.2016.9.1447 Hynd, 2019, Lipschitz regularity for a homogeneous doubly nonlinear PDE, SIAM J. Math. Anal., 51, 3606, 10.1137/19M1246201 Iagar, 2008, Radial equivalence for the two basic nonlinear degenerate diffusion equations, J. Math. Pures Appl. Neuvième Série, 89, 1, 10.1016/j.matpur.2007.09.002 Ishige, 1996, On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation, SIAM J. Math. Anal., 27, 1235, 10.1137/S0036141094270370 Ivanov, 1994, Regularity for doubly nonlinear parabolic equations, vol. 209, 37 Juutinen, 2009, Pointwise decay for the solutions of degenerate and singular parabolic equations, Adv. Differ. Equ., 14, 663 Kalashnikov, 1987, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russ. Math. Surv., 42, 169, 10.1070/RM1987v042n02ABEH001309 Kamin, 1988, Fundamental solutions and asymptotic behaviour for the p-Laplacian equations, Rev. Mat. Iberoam., 4, 339, 10.4171/RMI/77 King, 1990, Exact similarity solutions to some nonlinear diffusion equations, J. Phys. A, Math. Gen., 23, 3681, 10.1088/0305-4470/23/16/017 Kinnunen, 2007, Local behaviour of solutions to doubly nonlinear parabolic equations, Math. Ann., 337, 705, 10.1007/s00208-006-0053-3 Komura, 1967, Nonlinear semigroups in Hilbert space, J. Math. Soc. Jpn., 19, 493, 10.2969/jmsj/01940493 Kuusi, 2012, Pointwise gradient estimates, Nonlinear Anal., Theory Methods Appl., 75, 4650, 10.1016/j.na.2011.11.021 Kuusi, 2013, Gradient regularity for nonlinear parabolic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 12, 755 Kuusi, 2014, Borderline gradient continuity for nonlinear parabolic systems, Math. Ann., 360, 937, 10.1007/s00208-014-1055-1 Kuusi, 2014, Guide to nonlinear potential estimates, Bull. Math. Sci., 4, 1, 10.1007/s13373-013-0048-9 Kuusi, 2014, A boundary Harnack inequality for singular equations of p-parabolic type, Proc. Am. Math. Soc., 142, 2705, 10.1090/S0002-9939-2014-12171-X Kuusi, 2012, Local Hölder continuity for doubly nonlinear parabolic equations, Indiana Univ. Math. J., 61, 399, 10.1512/iumj.2012.61.4513 Ladyzhenskaya, 1969, The mathematical theory of viscous incompressible flow, vol. 2 Manfredi, 2010, An asymptotic mean value characterization for p-harmonic functions, Proc. Am. Math. Soc., 138, 881, 10.1090/S0002-9939-09-10183-1 Manfredi, 1994, Large time behaviour of solutions to a class of doubly nonlinear parabolic equations, Electron. J. Differ. Equ. Moser, 1964, A Harnack inequality for parabolic differential equations, Commun. Pure Appl. Math., 17, 101, 10.1002/cpa.3160170106 Nirenberg, 1959, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3), 13, 115 Nirenberg, 1966, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3), 20, 733 Peres, 2008, Tug-of-war with noise: a game-theoretic view of the p-Laplacian, Duke Math. J., 145, 91, 10.1215/00127094-2008-048 Pini, 1954, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Semin. Mat. Univ. Padova, Math. J. Univ. Padova, 23, 422 Porzio, 1993, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equ., 103, 146, 10.1006/jdeq.1993.1045 Recalde, 2016, Backward estimates for nonnegative solutions to a class of singular parabolic equations, Nonlinear Anal., Theory Methods Appl., 144, 194, 10.1016/j.na.2016.06.013 Savaré, 1994, The asymptotic profile of solutions of a class of doubly nonlinear equations, Nonlinear Anal., Theory Methods Appl., 22, 1553, 10.1016/0362-546X(94)90188-0 Simon, 1986, Compact sets in the space Lp(0,T;B), Ann. Mat. Pura Appl., 146, 65, 10.1007/BF01762360 Simonov, 2020 Stan, 2018, Porous medium equation with nonlocal pressure, 277 Stan, 2014, The Fisher-KPP equation with nonlinear fractional diffusion, SIAM J. Math. Anal., 46, 3241, 10.1137/130918289 Vázquez, 2020, The evolution fractional p-Laplacian equation in RN. Fundamental solution and asymptotic behaviour, Nonlinear Anal., Theory Methods Appl., 199, 10.1016/j.na.2020.112034 Vázquez, 2003, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3, 67, 10.1007/s000280300004 Vázquez, 2006, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, vol. 33 Vázquez, 2007, The Porous Medium Equation Vázquez, 2017, The mathematical theories of diffusion: nonlinear and fractional diffusion, 205 Vázquez Vázquez, 2021, The fractional p-Laplacian evolution equation in Rn in the sublinear case, Calc. Var. Partial Differ. Equ., 60, 10.1007/s00526-021-02005-6 Véron, 1979, Effets régularisants de semi-groupes non linéaires dans des espaces de Banach, Ann. Fac. Sci. Toulouse: Math., 1, 171, 10.5802/afst.535 Widder, 1975, The Heat Equation, vol. 67 Wu, 2001 Zhao, 1995, The Cauchy problem for ut=div(|∇u|p−2∇u) when 2NN+1<p<2, Nonlinear Anal., Theory Methods Appl., 24, 615, 10.1016/0362-546X(94)00110-4