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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 |
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. |
show all references
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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