Wellposedness of the Dirichlet problem for the nonlinear diffusion equation in nonsmooth domains
Author:
Ugur G. Abdulla
Translated by:
Journal:
Trans. Amer. Math. Soc. 357 (2005), 247265
MSC (2000):
Primary 35K65, 35K55
Published electronically:
February 27, 2004
MathSciNet review:
2098094
Fulltext PDF Free Access
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Abstract: We investigate the Dirichlet problem for the parablic equation
in a nonsmooth domain . In a recent paper [U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384403] 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 initialboundary data. In particular, we prove contraction estimation in general nonsmooth domains.
 1.
U.G. Abdulla, On the Dirichlet problem for the nonlinear diffusion equation in nonsmooth domains, Journal of Mathematical Analysis and Applications, 260, 2 (2001), 384403. 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), 14221434.
 3.
U.G. Abdulla, Reactiondiffusion in irregular domains, Journal of Differential Equations, 164 (2000), 321354. MR 2001d:35094
 4.
U.G. Abdulla, Reactiondiffusion in a closed domain formed by irregular curves, Journal of Mathematical Analysis and Applications, 246 (2000), 480492. MR 2001f:35194
 5.
U.G. Abdulla and J.R. King, Interface development and local solutions to reactiondiffusion equations, SIAM Journal of Math. Anal., 32, No. 2 (2000), 235260. 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), 541560. MR 2003g:35116
 7.
H.W. Alt and S. Luckhaus, Quasilinear ellipticparabolic differential equations, Math. Z., 183 (1983), 311341. MR 85c:35059
 8.
D. G. Aronson, The porous media equation, in ``Nonlinear Diffusion Problems" (A. Fasano and M. Primicerio, eds.), pp. 146, Lecture Notes in Mathematics, Vol. 1224, SpringerVerlag, 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), 378412. 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), 99113. MR 80i:35090
 11.
E. DiBenedetto, Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Para Appl., (4) CXXX (1982), 131176. MR 83k:35045
 12.
E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83118. MR 85c:35010
 13.
A. Friedman, Partial Differential Equations of Parabolic Type, PrenticeHall, 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), 659675. MR 83h:35061
 15.
A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate secondorder parabolic equations, Russian Math. Surveys, 42 (1987), 169222. 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), 387409. 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. 347389, 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), 733748. MR 83e:35074
 1.
 U.G. Abdulla, On the Dirichlet problem for the nonlinear diffusion equation in nonsmooth domains, Journal of Mathematical Analysis and Applications, 260, 2 (2001), 384403. 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), 14221434.
 3.
 U.G. Abdulla, Reactiondiffusion in irregular domains, Journal of Differential Equations, 164 (2000), 321354. MR 2001d:35094
 4.
 U.G. Abdulla, Reactiondiffusion in a closed domain formed by irregular curves, Journal of Mathematical Analysis and Applications, 246 (2000), 480492. MR 2001f:35194
 5.
 U.G. Abdulla and J.R. King, Interface development and local solutions to reactiondiffusion equations, SIAM Journal of Math. Anal., 32, No. 2 (2000), 235260. 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), 541560. MR 2003g:35116
 7.
 H.W. Alt and S. Luckhaus, Quasilinear ellipticparabolic differential equations, Math. Z., 183 (1983), 311341. MR 85c:35059
 8.
 D. G. Aronson, The porous media equation, in ``Nonlinear Diffusion Problems" (A. Fasano and M. Primicerio, eds.), pp. 146, Lecture Notes in Mathematics, Vol. 1224, SpringerVerlag, 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), 378412. 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), 99113. MR 80i:35090
 11.
 E. DiBenedetto, Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Para Appl., (4) CXXX (1982), 131176. MR 83k:35045
 12.
 E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83118. MR 85c:35010
 13.
 A. Friedman, Partial Differential Equations of Parabolic Type, PrenticeHall, 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), 659675. MR 83h:35061
 15.
 A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate secondorder parabolic equations, Russian Math. Surveys, 42 (1987), 169222. 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), 387409. 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. 347389, 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), 733748. 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 329016975
Email:
abdulla@fit.edu
DOI:
http://dx.doi.org/10.1090/S0002994704034646
PII:
S 00029947(04)034646
Keywords:
Dirichlet problem,
nonsmooth domains,
nonlinear 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
