Skip to main content
Boundary Value Problems
Download PDF

Reaction-Diffusion in Nonsmooth and Closed Domains

Boundary Value Problems volume 2007, Article number: 031261 (2006) Cite this article

Abstract

We investigate the Dirichlet problem for the parabolic equation in a nonsmooth and closed domain possibly formed with irregular surfaces and having a characteristic vertex point. Existence, boundary regularity, uniqueness, and comparison results are established. The main objective of the paper is to express the criteria for the well-posedness in terms of the local modulus of lower semicontinuity of the boundary manifold. The two key problems in that context are the boundary regularity of the weak solution and the question whether any weak solution is at the same time a viscosity solution.

[12345678910111213141516171819]

References

  1. Kalashnikov AS: Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations. Russian Mathematical Surveys 1987,42(2):169–222. 10.1070/RM1987v042n02ABEH001309

    MATH  MathSciNet  Article  Google Scholar 

  2. Abdulla UG: On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains. Journal of Mathematical Analysis and Applications 2001,260(2):384–403. 10.1006/jmaa.2001.7458

    MATH  MathSciNet  Article  Google Scholar 

  3. Abdulla UG: Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains. Transactions of the American Mathematical Society 2005,357(1):247–265. 10.1090/S0002-9947-04-03464-6

    MATH  MathSciNet  Article  Google Scholar 

  4. DiBenedetto E: Continuity of weak solutions to certain singular parabolic equations. Annali di Matematica Pura ed Applicata. Serie Quarta 1982,130(1):131–176. 10.1007/BF01761493

    MATH  MathSciNet  Article  Google Scholar 

  5. DiBenedetto E: Continuity of weak solutions to a general porous medium equation. Indiana University Mathematics Journal 1983,32(1):83–118. 10.1512/iumj.1983.32.32008

    MATH  MathSciNet  Article  Google Scholar 

  6. Ziemer WP: Interior and boundary continuity of weak solutions of degenerate parabolic equations. Transactions of the American Mathematical Society 1982,271(2):733–748. 10.1090/S0002-9947-1982-0654859-7

    MATH  MathSciNet  Article  Google Scholar 

  7. Abdulla UG: On the Dirichlet problem for reaction-diffusion equations in non-smooth domains. Nonlinear Analysis 2001,47(2):765–776. 10.1016/S0362-546X(01)00221-8

    MATH  MathSciNet  Article  Google Scholar 

  8. Abdulla UG: First boundary value problem for the diffusion equation. I. Iterated logarithm test for the boundary regularity and solvability. SIAM Journal on Mathematical Analysis 2003,34(6):1422–1434. 10.1137/S0036141002415049

    MATH  MathSciNet  Article  Google Scholar 

  9. Abdulla UG: Multidimensional Kolmogorov-Petrovsky test for the boundary regularity and irregularity of solutions to the heat equation. Boundary Value Problems 2005,2005(2):181–199. 10.1155/BVP.2005.181

    MATH  MathSciNet  Article  Google Scholar 

  10. Petrovsky IG: Zur ersten Randwertaufgabe der Wärmeleitungsgleichung. Compositio Mathematica 1935, 1: 383–419.

    MathSciNet  Google Scholar 

  11. Abdulla UG: Reaction-diffusion in a closed domain formed by irregular curves. Journal of Mathematical Analysis and Applications 2000,246(2):480–492. 10.1006/jmaa.2000.6800

    MATH  MathSciNet  Article  Google Scholar 

  12. Abdulla UG: Reaction-diffusion in irregular domains. Journal of Differential Equations 2000,164(2):321–354. 10.1006/jdeq.2000.3761

    MATH  MathSciNet  Article  Google Scholar 

  13. Abdulla UG: Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption. Nonlinear Analysis 2002,50(4):541–560. 10.1016/S0362-546X(01)00764-7

    MATH  MathSciNet  Article  Google Scholar 

  14. Abdulla UG, King JR: Interface development and local solutions to reaction-diffusion equations. SIAM Journal on Mathematical Analysis 2000,32(2):235–260. 10.1137/S003614109732986X

    MATH  MathSciNet  Article  Google Scholar 

  15. Wiener N: The Dirichlet problem. Journal of Mathematics and Physics 1924, 3: 127–146.

    MATH  Google Scholar 

  16. Feireisl E, Petzeltová H, Simondon F: Admissible solutions for a class of nonlinear parabolic problems with non-negative data. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2001,131(4):857–883. 10.1017/S0308210500001153

    MATH  MathSciNet  Article  Google Scholar 

  17. Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall, New Jersey; 1964:xiv+347.

    MATH  Google Scholar 

  18. Ladyzhenskaya OA, Solonnikov VA, Uralceva NN: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Rhode Island; 1968.

    Google Scholar 

  19. Lieberman GM: Second Order Parabolic Differential Equations. World Scientific, New Jersey; 1996:xii+439.

    MATH  Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL, 32901, USA

    Ugur G Abdulla

Authors
  1. Ugur G Abdulla
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ugur G Abdulla.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Abdulla, U.G. Reaction-Diffusion in Nonsmooth and Closed Domains. Bound Value Probl 2007, 031261 (2006). https://doi.org/10.1155/2007/31261

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/31261

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Weak Solution
  • Functional Equation