Advanced Search
Article Contents
Article Contents

On the local behavior of non-negative solutions to a logarithmically singular equation

Abstract Related Papers Cited by
  • The local positivity of solutions to logarithmically singular diffusion equations is investigated in some open space-time domain $E\times(0,T]$. It is shown that if at some time level $t_o\in(0,T]$ and some point $x_o\in E$ the solution $u(\cdot,t_o)$ is not identically zero in a neighborhood of $x_o$, in a measure-theoretical sense, then it is strictly positive in a neighborhood of $(x_o, t_o)$. The precise form of this statement is by an intrinsic Harnack-type inequality, which also determines the size of such a neighborhood.
    Mathematics Subject Classification: Primary: 35K65, 35B65; Secondary: 35B45.


    \begin{equation} \\ \end{equation}
  • [1]

    M. Bonforte and J. L. Vázquez, Positivity, local smoothing and Harnack inequalities for very fast diffusion equations, Adv. Math., 223 (2010), 529-578.doi: 10.1016/j.aim.2009.08.021.


    J. P. Burelbach, S. G. Bankoff and S. H. Davis, Nonlinear stability of evaporating/condensing liquid films, J. Fluid Mech., 195 (1988), 463-494.doi: 10.1017/S0022112088002484.


    S.-C. Chang, S.-K. Hong and C.-T. Wu, The Harnack estimate for the modified Ricci flow on complete $\mathbb R^2$, Rocky Mountain J. of Math., 33 (2003), 69-92.doi: 10.1216/rmjm/1181069987.


    J. T. Chayes, S. J. Osher and J. V. Ralston, On singular diffusion equations with applications to self-organized criticality, Comm. Pure Appl. Math., 46 (1993), 1363-1377.doi: 10.1002/cpa.3160461004.


    S. H. Davis, E. DiBenedetto and D. J. Diller, Some a priori estimates for a singular evolution equation arising in thin-film dynamics, SIAM J. Math. Anal., 27 (1996), 638-660.doi: 10.1137/0527035.


    P. Daskalopoulos and M. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Rational Mech. Anal., 137(4), (1997), 363-380.


    P. Daskalopoulos and M. del Pino, On the Cauchy problem for $u_t=\Delta\log u$ in higher dimensions, Math. Ann., 313 (1999), 189-206.doi: 10.1007/s002080050257.


    P. Daskalopoulos and M. Del Pino, Nonradial solvability structure of super-diffusive nonlinear parabolic equations, Trans. Amer. Math. Soc., 354 (2002), 1583-1599.doi: 10.1090/S0002-9947-01-02888-4.


    P. G. de Gennes, Wetting: Statics and dynamics, Rev. Modern Phys., 57 (1985), 827-863.doi: 10.1103/RevModPhys.57.827.


    E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer-Verlag, New York, 1993.


    E. DiBenedetto and D. J. Diller, About a singular parabolic equation arising in thin film dynamics and in the Ricci flow for complete $\mathbb R^2$, in "Partial Differential Equations and Applications," Lecture Notes in Pure and Appl. Math., 177, Dekker, New York, (1996), 103-119.


    E. DiBenedetto, U. Gianazza and V. Vespri, Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations, Manuscripta Mathematica, 131 (2010), 231-245.doi: 10.1007/s00229-009-0317-9.


    E. DiBenedetto, U. Gianazza and V. Vespri, "Harnack's Inequality for Degenerate and Singular Parabolic Equations," Springer Monographs in Mathematics, Springer-Verlag, New York, 2012.


    J. R. Esteban, A. Rodríguez and J. L. Vázquez, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations, 139 (1988), 985-1039.doi: 10.1080/03605308808820566.


    R. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom., 37 (1993), 225-243.


    K. M. Hui, Singular limit of solutions of the equation $u_t=\Delta\frac{u^m}m$ as $m\rightarrow0$, Pacific J. Math., 187 (1999), 297-316.doi: 10.2140/pjm.1999.187.297.


    K. M. Hui, Singular limit of solutions of the very fast diffusion equation, Nonlinear Anal., 68 (2008), 1120-1147.doi: 10.1016/j.na.2006.12.009.


    H. P. McKean, The central limit theorem for Carleman's equation, Israel J. Math., 21 (1975), 54-92.doi: 10.1007/BF02757134.


    P. Rosenau, Fast and superfast diffusion processes, Physical Rev. Lett., 74 (1995), 1056-1059.doi: 10.1103/PhysRevLett.74.1056.


    J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl. (9), 71 (1992), 503-526.


    J. L. Vázquez, Failure of the strong maxmum principle in nonlinear diffusion. Existence of needles, Comm. Partial Differential Equations, 30 (2005), 1263-1303.doi: 10.1080/10623320500258759.


    M. B. Williams and S. H. Davis, Nonlinear theory of film rupture, Jour. of Colloid and Interface Sc., 90 (1982), 220-228.doi: 10.1016/0021-9797(82)90415-5.


    L.-F. Wu, The Ricci flow on complete $\mathbb R^2$, Comm. in Anal. Geom., 1 (1993), 439-472.

  • 加载中

Article Metrics

HTML views() PDF downloads(59) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint