Nonradial solvability structure of super-diffusive nonlinear parabolic equations
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- by Panagiota Daskalopoulos and Manuel del Pino
- Trans. Amer. Math. Soc. 354 (2002), 1583-1599
- DOI: https://doi.org/10.1090/S0002-9947-01-02888-4
- Published electronically: December 4, 2001
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Abstract:
We study the solvability of the Cauchy problem for the nonlinear parabolic equation \[ \frac {\partial u}{\partial t} = \mbox {div} (u^{m-1}\nabla u)\] when $m < 0$ in $\textbf {R}^2$, with $u(x,0)= f(x)$ a given nonnegative function. It is known from earlier works of the authors that the asymptotic radial growth $r^{-2/1-m}$, $r=|x|$ for the spherical averages of $f(x)$ is critical for local solvability, in particular ensuring it if $f$ is radially symmetric. We show that if the initial data $f(x)$ behaves in polar coordinates like $r^{-2/1-m} g(\theta )$, for large $r= |x|$ with $g$ nonnegative and $2\pi$-periodic, then the following holds: If $g$ vanishes on some interval of length $l^* = \frac {(m-1)\pi }{2m} >0$, then there is no local solution of the initial value problem. On the other hand, if such an interval does not exist, then the initial value problem is locally solvable and the time of existence can be estimated explicitly.References
- Donald G. Aronson and Philippe Bénilan, Régularité des solutions de l’équation des milieux poreux dans $\textbf {R}^{N}$, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 2, A103–A105 (French, with English summary). MR 524760
- D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), no. 1, 351–366. MR 712265, DOI 10.1090/S0002-9947-1983-0712265-1
- Philippe Bénilan, Michael G. Crandall, and Michel Pierre, Solutions of the porous medium equation in $\textbf {R}^{N}$ under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), no. 1, 51–87. MR 726106, DOI 10.1512/iumj.1984.33.33003
- 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), no. 10, 1363–1377. MR 1234338, DOI 10.1002/cpa.3160461004
- M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), no. 2, 193–222. MR 427826, DOI 10.1080/03605307708820029
- Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions to fast diffusions, Rev. Mat. Iberoamericana 4 (1988), no. 1, 11–29. MR 1009117, DOI 10.4171/RMI/61
- Panagiota Daskalopoulos and Manuel A. del Pino, On fast diffusion nonlinear heat equations and a related singular elliptic problem, Indiana Univ. Math. J. 43 (1994), no. 2, 703–728. MR 1291536, DOI 10.1512/iumj.1994.43.43030
- Panagiota Daskalopoulos and Manuel Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Rational Mech. Anal. 137 (1997), no. 4, 363–380. MR 1463800, DOI 10.1007/s002050050033
- Panagiota Daskalopoulos and Manuel A. del Pino, On a singular diffusion equation, Comm. Anal. Geom. 3 (1995), no. 3-4, 523–542. MR 1371208, DOI 10.4310/CAG.1995.v3.n3.a5
- Stephen H. Davis, Emmanuele DiBenedetto, and David J. Diller, Some a priori estimates for a singular evolution equation arising in thin-film dynamics, SIAM J. Math. Anal. 27 (1996), no. 3, 638–660. MR 1382826, DOI 10.1137/0527035
- Juan R. Esteban, Ana Rodríguez, and Juan L. Vázquez, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations 13 (1988), no. 8, 985–1039. MR 944437, DOI 10.1080/03605308808820566
- de Gennes, P.G., Wetting: statics and dynamics, Reviews of Modern Physics, 57 No 3, 1985, pp. 827-863.
- de Gennes, P.G., Spreading laws for microscopic droplets, C.R. Acad. Sci., Paris II, 298, 1984, pp. 475-478.
- Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262. MR 954419, DOI 10.1090/conm/071/954419
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
- Lonngren, K.E., Hirose, A., Expansion of an electron cloud, Phys. Lett. A, 59, 1976, pp 285-286.
- Lopez, J., Miller, C.A., Ruckenstein, E., Spreading kinetics of liquid drops on solids, J. Coll. Int. Sci. 56, 1976, pp 460-461.
- Rosenau, P., On a Fast and Super-Fast Diffusion, preprint.
- Stratov, V.M., Speading of droplets of nonvotiable liquids over flat solid surface, Coll. J. USSR 45, 1983, 1009-1014.
- Juan Luis Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl. (9) 71 (1992), no. 6, 503–526. MR 1193606
- Juan L. Vázquez, Juan R. Esteban, and Ana Rodríguez, The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane, Adv. Differential Equations 1 (1996), no. 1, 21–50. MR 1357953
- Lang-Fang Wu, A new result for the porous medium equation derived from the Ricci flow, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 1, 90–94. MR 1164949, DOI 10.1090/S0273-0979-1993-00336-7
Bibliographic Information
- Panagiota Daskalopoulos
- Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697
- MR Author ID: 353551
- Email: pdaskalo@math.uci.edu
- Manuel del Pino
- Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
- MR Author ID: 56185
- Email: delpino@dim.uchile.cl
- Received by editor(s): March 25, 1998
- Published electronically: December 4, 2001
- Additional Notes: The first author was partially supported by The Sloan Foundation and by NSF/Conicyt-Chile grant INT-9802406
The second author was partially supported by grants Lineas Complementarias Fondecyt 8000010 and FONDAP - © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 1583-1599
- MSC (1991): Primary 35K15, 35K55, 35K65; Secondary 35J40
- DOI: https://doi.org/10.1090/S0002-9947-01-02888-4
- MathSciNet review: 1873019