American Institute of Mathematical Sciences

September  2012, 17(6): 1841-1858. doi: 10.3934/dcdsb.2012.17.1841

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

 1 Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240, United States, United States 2 Dipartimento di Matematica "F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia

Received  September 2011 Revised  November 2011 Published  May 2012

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.
Citation: Emmanuele DiBenedetto, Ugo Gianazza, Naian Liao. On the local behavior of non-negative solutions to a logarithmically singular equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1841-1858. doi: 10.3934/dcdsb.2012.17.1841
References:
 [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. [2] 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. [3] 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. [4] 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. [5] 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. [6] P. Daskalopoulos and M. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Rational Mech. Anal., 137(4), (1997), 363-380. [7] 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. [8] 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. [9] P. G. de Gennes, Wetting: Statics and dynamics, Rev. Modern Phys., 57 (1985), 827-863. doi: 10.1103/RevModPhys.57.827. [10] E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer-Verlag, New York, 1993. [11] 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. [12] 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. [13] 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. [14] 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. [15] R. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom., 37 (1993), 225-243. [16] 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. [17] 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. [18] H. P. McKean, The central limit theorem for Carleman's equation, Israel J. Math., 21 (1975), 54-92. doi: 10.1007/BF02757134. [19] P. Rosenau, Fast and superfast diffusion processes, Physical Rev. Lett., 74 (1995), 1056-1059. doi: 10.1103/PhysRevLett.74.1056. [20] J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl. (9), 71 (1992), 503-526. [21] 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. [22] 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. [23] L.-F. Wu, The Ricci flow on complete $\mathbb R^2$, Comm. in Anal. Geom., 1 (1993), 439-472.

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References:
 [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. [2] 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. [3] 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. [4] 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. [5] 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. [6] P. Daskalopoulos and M. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Rational Mech. Anal., 137(4), (1997), 363-380. [7] 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. [8] 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. [9] P. G. de Gennes, Wetting: Statics and dynamics, Rev. Modern Phys., 57 (1985), 827-863. doi: 10.1103/RevModPhys.57.827. [10] E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer-Verlag, New York, 1993. [11] 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. [12] 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. [13] 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. [14] 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. [15] R. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom., 37 (1993), 225-243. [16] 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. [17] 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. [18] H. P. McKean, The central limit theorem for Carleman's equation, Israel J. Math., 21 (1975), 54-92. doi: 10.1007/BF02757134. [19] P. Rosenau, Fast and superfast diffusion processes, Physical Rev. Lett., 74 (1995), 1056-1059. doi: 10.1103/PhysRevLett.74.1056. [20] J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl. (9), 71 (1992), 503-526. [21] 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. [22] 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. [23] L.-F. Wu, The Ricci flow on complete $\mathbb R^2$, Comm. in Anal. Geom., 1 (1993), 439-472.
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