Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Nonradial solvability structure of super-diffusive nonlinear parabolic equations

Authors: Panagiota Daskalopoulos and Manuel del Pino
Journal: Trans. Amer. Math. Soc. 354 (2002), 1583-1599
MSC (1991): Primary 35K15, 35K55, 35K65; Secondary 35J40
Published electronically: December 4, 2001
MathSciNet review: 1873019
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the solvability of the Cauchy problem for the nonlinear parabolic equation

\begin{displaymath}\frac {\partial u}{\partial t} = \mbox{div}\, (u^{m-1}\nabla u)\end{displaymath}

when $m < 0$ in ${\bf R}^2$, with $u(x,0)= f(x)$ a given nonnegative function. It is known from earlier works of the authors that the asymptotic radial growth $r^{-2/1-m}$, $r=\vert x\vert$ 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 $r= \vert x\vert$ 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 [Enhancements On Off] (What's this?)

  • 1. Aronson, D.G., Bénilan P., Régularité des solutions de l'équation de milieux poreux dans ${\bf R}^n$, C.R. Acad. Sci. Paris, 288, 1979, pp 103-105. MR 82i:35090
  • 2. Aronson, D.G. and Caffarelli, L.A., The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc., 280, 1983, pp. 351-366. MR 85c:35042
  • 3. Bénilan, P., Crandall, M.G. and Pierre, M., Solutions of the porous medium equation under optimal conditions on the initial values, Indiana Univ. Math.J., 33, 1984, pp. 51-87. MR 86b:35084
  • 4. Chayes, J.T., Osher, S.J. and Ralston, J.V., On singular diffusion equations with applications to self-organized criticality, Comm. Pure Appl. Math. 46, 1993, pp. 1363-1377. MR 94i:35086
  • 5. Crandall, M.G., Rabinowitz, P.H., Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. Partial Diff. Eq., 2, 1977, pp. 193-222. MR 55:856
  • 6. Dahlberg, B.E.G., Kenig, C.E., Non-negative solutions to fast diffusions, Rev. Mat. Iberoamericana, 4, 1988, pp. 11-29. MR 90i:35128
  • 7. Daskalopoulos, P., del Pino M.A., On fast diffusion nonlinear heat equations and a related singular elliptic problem, Indiana Univ. Math. J., 43, 1994, pp. 703-728. MR 95g:35078
  • 8. Daskalopoulos,P., del Pino M.A., On nonlinear parabolic equations of very fast diffusion, Arch. Rational Mech. Analysis, 137, 1997, pp. 363-380. MR 98d:35094
  • 9. Daskalopoulos,P., del Pino M.A., On a Singular Diffusion Equation, Comm. in Analysis and Geometry, Vol. 3, 1995, pp 523-542. MR 97b:35116
  • 10. Davis, S.H., DiBenedetto, E., Diller, D.,J., Some a priori estimates for a singular evolution equation arising in thin film dynamics, SIAM J. Math. Anal., 27, 1996, pp. 638-660. MR 97a:35095
  • 11. Esteban, J.R., Rodriguez, A., Vazquez, J.L., A nonlinear heat equation with singular diffusivity, Arch. Rational Mech. Analysis, 103, 1988, pp. 985-1039. MR 89h:35167
  • 12. de Gennes, P.G., Wetting: statics and dynamics, Reviews of Modern Physics, 57 No 3, 1985, pp. 827-863.
  • 13. de Gennes, P.G., Spreading laws for microscopic droplets, C.R. Acad. Sci., Paris II, 298, 1984, pp. 475-478.
  • 14. Hamilton, R., The Ricci flow on surfaces, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237-262. MR 89i:53029
  • 15. Ladyzenskaja, O.A., Solonnikov, V.A., and Ural'ceva, N.N., Linear and quasilinear equations of parabolic type, Translations of Mathematical Monogrphs, 23, 1967. MR 39:3159b
  • 16. Lonngren, K.E., Hirose, A., Expansion of an electron cloud, Phys. Lett. A, 59, 1976, pp 285-286.
  • 17. Lopez, J., Miller, C.A., Ruckenstein, E., Spreading kinetics of liquid drops on solids, J. Coll. Int. Sci. 56, 1976, pp 460-461.
  • 18. Rosenau, P., On a Fast and Super-Fast Diffusion, preprint.
  • 19. Stratov, V.M., Speading of droplets of nonvotiable liquids over flat solid surface, Coll. J. USSR 45, 1983, 1009-1014.
  • 20. Vázquez, J.L., Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl., 71, 1992, pp. 503-526. MR 93k:35133
  • 21. Vázquez, J.L., Esteban, J.R., Rodriguez, R., The fast diffusion with logarithmic nonlinearity and the evolution of conformal metrics in the plane, Advances in Diff. Equations, 1, 1996. MR 97b:35096
  • 22. Wu, L.-F., A new result for the porous medium equation derived from the Ricci flow, Bull. Amer. Math. Soc., 28, 1993, pp 90-94. MR 93f:58245

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35K15, 35K55, 35K65, 35J40

Retrieve articles in all journals with MSC (1991): 35K15, 35K55, 35K65, 35J40

Additional Information

Panagiota Daskalopoulos
Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697

Manuel del Pino
Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

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
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society