×

Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations. (English) Zbl 1184.35083

The authors investigate qualitative properties of local solutions \(u(t,x)\geqslant 0\) to the fast diffusion equation, \(\partial _tu=\Delta (u^m)/m\) with \(m<1\). For such a kind of equation it is known that intrinsic Harnack inequality does not hold for low \(m\) in the so-called very fast diffusion range, precisely for all \(m\leqslant m_c=(d - 2)/d\). Their main results are quantitative positivity and boundedness estimates for locally defined solutions that they combine into forms of interesting new Harnack-like inequalities. The boundedness statements are true even for \(m\leqslant 0\).

MSC:

35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35K59 Quasilinear parabolic equations
35K65 Degenerate parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aronson, D. G.; Caffarelli, L. A., The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc., 280, 351-366 (1983) · Zbl 0556.76084
[2] Aronson, D. G.; Peletier, L. A., Large time behaviour of solutions of the porous medium equation in bounded domains, J. Differential Equations, 39, 378-412 (1981) · Zbl 0475.35059
[3] Bénilan, P.; Crandall, M. G., The continuous dependence on \(φ\) of solutions of \(u_t - \Delta \varphi(u) = 0\), Indiana Univ. Math. J., 30, 161-177 (1981) · Zbl 0482.35012
[4] Bénilan, P.; Crandall, M. G., Regularizing effects of homogeneous evolution equations, (Contributions to Analysis and Geometry (suppl. to Amer. J. Math.) (1981), Johns Hopkins Univ. Press: Johns Hopkins Univ. Press Baltimore, MD), 23-39
[5] Bertsch, M.; Dal Passo, R.; Ughi, M., Discontinuous “viscosity” solutions of a degenerate parabolic equation, Trans. Amer. Math. Soc., 320, 779-798 (1990) · Zbl 0714.35039
[6] Bertsch, M.; Ughi, M., Positivity properties of viscosity solutions of a degenerate parabolic equation, Nonlinear Anal., 14, 7, 571-592 (1990) · Zbl 0702.35044
[7] Bonforte, M.; Grillo, G.; Vázquez, J. L., Fast diffusion flow on manifolds of nonpositive curvature, J. Evol. Equ., 8, 1, 99-128 (2008) · Zbl 1139.35065
[8] Bonforte, M.; Vázquez, J. L., Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., 240, 2, 399-428 (2006) · Zbl 1107.35063
[11] Brezis, H.; Friedman, A., Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. (9), 62, 1, 73-97 (1983) · Zbl 0527.35043
[12] Chasseigne, E.; Vazquez, J. L., Theory of extended solutions for fast diffusion equations in optimal classes of data. Radiation from singularities, Arch. Ration. Mech. Anal., 164, 133-187 (2002) · Zbl 1018.35048
[13] Daskalopoulos, P.; del Pino, M., On nonlinear parabolic equations of very fast diffusion, Arch. Ration. Mech. Anal., 137, 4, 363-380 (1997) · Zbl 0886.35081
[14] Daskalopoulos, P.; Kenig, C. E., Degenerate Diffusions. Initial Value Problems and Local Regularity Theory, EMS Tracts Math., vol. 1 (2007), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich · Zbl 1205.35002
[15] Del Pino, M.; Saez, M., On the extinction profile for solutions of \(u_t = \Delta u^{(N - 2) /(N + 2)}\), Indiana Univ. Math. J., 50, 1, 611-628 (2001) · Zbl 0991.35011
[16] Diaz, G.; Diaz, J. I., Finite extinction time for a class of non-linear parabolic equations, Comm. Partial Differential Equations, 4, 1213-1231 (1979) · Zbl 0425.35057
[17] DiBenedetto, E., Degenerate Parabolic Equations, Universitext (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0794.35090
[19] DiBenedetto, E.; Kwong, Y. C., Harnack estimates and extinction profile for weak solution of certain singular parabolic equations, Trans. Amer. Math. Soc., 330, 2, 783-811 (1992) · Zbl 0772.35006
[20] DiBenedetto, E.; Kwong, Y. C.; Vespri, V., Local space-analyticity of solutions of certain singular parabolic equations, Indiana Univ. Math. J., 40, 2, 741-765 (1991) · Zbl 0784.35055
[21] DiBenedetto, E.; Urbano, M.; Vespri, V., Current issues on singular and degenerate evolution equations, (Evolutionary Equations, vol. I. Evolutionary Equations, vol. I, Handb. Differ. Equ. (2004), North-Holland: North-Holland Amsterdam), 169-286 · Zbl 1082.35002
[22] Galaktionov, V. A.; Vázquez, J. L., A Stability Technique for Evolution Partial Differential Equations, a Dynamical System Approach, Progr. Nonlinear Differential Equations Appl., vol. 56 (2004), Birkhäuser: Birkhäuser Boston
[23] Hamilton, R. S., The Ricci flow on surfaces, (Mathematics and General Relativity. Mathematics and General Relativity, Santa Cruz, CA, 1986. Mathematics and General Relativity. Mathematics and General Relativity, Santa Cruz, CA, 1986, Contemp. Math., vol. 71 (1988), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 237-262
[24] Herrero, M. A.; Pierre, M., The Cauchy problem for \(u_t = \Delta u^m\) when \(0 < m < 1\), Trans. Amer. Math. Soc., 291, 1, 145-158 (1985) · Zbl 0583.35052
[25] King, J. R., Self-similar behaviour for the equation of fast nonlinear diffusion, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 343, 337-375 (1993) · Zbl 0797.35097
[26] Moser, J., A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17, 101-134 (1964) · Zbl 0149.06902
[27] Pierre, M., Nonlinear fast diffusion with measures as data, (Nonlinear Parabolic Equations: Qualitative Properties of Solutions. Nonlinear Parabolic Equations: Qualitative Properties of Solutions, Rome, 1985. Nonlinear Parabolic Equations: Qualitative Properties of Solutions. Nonlinear Parabolic Equations: Qualitative Properties of Solutions, Rome, 1985, Pitman Res. Notes Math. Ser., vol. 149 (1987), Longman Sci. Tech.: Longman Sci. Tech. Harlow), 179-188
[28] Trudinger, N. S., On Harnack type inequalities and their applications to quasi linear parabolic equations, Comm. Pure Appl. Math., 21, 205-226 (1968) · Zbl 0159.39303
[29] Vázquez, J. L., Nonexistence of solutions for nonlinear heat equation of fast diffusion type, J. Math. Pures Appl., 71, 503-526 (1992) · Zbl 0694.35088
[30] Vázquez, J. L., Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Ser. Math. Appl., vol. 33 (2006), Oxford University Press
[31] Vázquez, J. L., The Porous Medium Equation. Mathematical Theory, Oxford Math. Monogr. (2007), Oxford University Press: Oxford University Press Oxford
[32] Vázquez, J. L., Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion, Discrete Contin. Dyn. Syst. Ser. B, 19, 1, 1-35 (2007)
[33] Widder, D. V., Positive temperatures on an infinite rod, Trans. Amer. Math. Soc., 55, 85-95 (1944) · Zbl 0061.22303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.