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Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains

Author: Ugur G. Abdulla
Translated by:
Journal: Trans. Amer. Math. Soc. 357 (2005), 247-265
MSC (2000): Primary 35K65, 35K55
Published electronically: February 27, 2004
MathSciNet review: 2098094
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Abstract: We investigate the Dirichlet problem for the parablic equation

\begin{displaymath}u_t = \Delta u^m, m > 0, \end{displaymath}

in a non-smooth domain $\Omega \subset \mathbb{R}^{N+1}, N \geq 2$. In a recent paper [U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove $L_1$-contraction estimation in general non-smooth domains.

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  • 1. U.G. Abdulla, On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, Journal of Mathematical Analysis and Applications, 260, 2 (2001), 384-403. MR 2002e:35122
  • 2. U.G. Abdulla, First boundary value problem for the diffusion equation. I. Iterated logarithm test for the boundary regularity and solvability, SIAM Journal of Math. Anal., 34, No. 6 (2003), 1422-1434.
  • 3. U.G. Abdulla, Reaction-diffusion in irregular domains, Journal of Differential Equations, 164 (2000), 321-354. MR 2001d:35094
  • 4. U.G. Abdulla, Reaction-diffusion in a closed domain formed by irregular curves, Journal of Mathematical Analysis and Applications, 246 (2000), 480-492. MR 2001f:35194
  • 5. U.G. Abdulla and J.R. King, Interface development and local solutions to reaction-diffusion equations, SIAM Journal of Math. Anal., 32, No. 2 (2000), 235-260. MR 2001g:35136
  • 6. U.G. Abdulla, Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption, Nonlinear Analysis, Theory, Methods and Applications, 50, 4 (2002), 541-560. MR 2003g:35116
  • 7. H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. MR 85c:35059
  • 8. D. G. Aronson, The porous media equation, in ``Nonlinear Diffusion Problems" (A. Fasano and M. Primicerio, eds.), pp. 1-46, Lecture Notes in Mathematics, Vol. 1224, Springer-Verlag, Berlin, 1986. MR 88a:35130
  • 9. D.G. Aronson and L.A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains, Journal of Differential Equations, 39 (1981), 378-412. MR 82g:35047
  • 10. L.A. Caffarelli and A. Friedman, Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc., 252 (1979), 99-113. MR 80i:35090
  • 11. E. DiBenedetto, Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Para Appl., (4) CXXX (1982), 131-176. MR 83k:35045
  • 12. E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83-118. MR 85c:35010
  • 13. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964. MR 31:6062
  • 14. B.H. Gilding and L.A. Peletier, Continuity of solutions of the porous media equation, Ann. Scuola Norm. Sup. Pisa, 8 (1981), 659-675. MR 83h:35061
  • 15. A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations, Russian Math. Surveys, 42 (1987), 169-222. MR 88h:35054
  • 16. O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence RI, 1968.
  • 17. G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. MR 98k:35003
  • 18. P.E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Analysis, TMA, 7 (1983), 387-409. MR 84d:35081
  • 19. J. L. Vazquez, An introduction to the mathematical theory of the porous medium equation, in ``Shape Optimization and Free Boundaries" (M. C. Delfour and G. Sabidussi, eds.), pp. 347-389, Kluwer Academic, Dordrecht, 1992. MR 95b:35101
  • 20. W.P. Ziemer, Interior and boundary continuity of weak solutions of degenerate parabolic equations, Transactions of the American Math. Soc., 271 (1982), 733-748. MR 83e:35074

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Additional Information

Ugur G. Abdulla
Affiliation: Department of Mathematical Sciences, Florida Institute of Technology, 150 West University Boulevard, Melbourne, Florida 32901-6975

Keywords: Dirichlet problem, non-smooth domains, non-linear diffusion, degenerate and singular parabolic equations, uniqueness and comparison results, $L_1$-contraction, boundary gradient estimates.
Received by editor(s): July 31, 2000
Received by editor(s) in revised form: July 21, 2003
Published electronically: February 27, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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