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Some Estimates of a System of Degenerate Parabolic Equations

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Abstract

The Dirichlet problem on a coupled system for degenerate parabolic equations is investigated in this paper. The aim is to show the existence, uniqueness of maximum solution, the continuity on the coupled functions, and time-dependent estimates.

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References

  1. Cockburn B, Gripenberg G. Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J Differ Equat. 1999;151:231–251.

    Article  MATH  MathSciNet  Google Scholar 

  2. Gilding BH. Properties of solutions of an equation in the theory of infiltration. Arch Ration Mech Anal. 1977;65:203–225.

    Article  MATH  MathSciNet  Google Scholar 

  3. Brandle C, Quiros F, Rossi JD. The role of non-linear diffusion in non-simultaneous blow-up. J Math Anal Appl. 2005;308:92–104.

    Article  MathSciNet  Google Scholar 

  4. Pao CV, Ruan WH. Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition. J Differ Equat. 2010;248:1175–1211.

    Article  MATH  MathSciNet  Google Scholar 

  5. Aronson DG, Benilan P. Régularité des solutions de l’équation des milieux poreux dans ℝℕ. CR Acad Sci Paris Sér A. 1979;288:103–105.

    MATH  MathSciNet  Google Scholar 

  6. Quirs F, Rossi J, Vazquez JL. Thermal avalanche for blow up solutions of semilinear heat equations. Commun Pur Appl Math. 2004;57:59–98.

    Article  Google Scholar 

  7. Friedman A. Partial differential equations of parabolic type. Englewood Cliffs: Prentice-Hall; 1964.

    MATH  Google Scholar 

  8. Galaktionov VA, Kurdymov P, Samarskii AA. On a parabolic system of quasilinear equations. Part I Differ Equat. 1983;19:2123–2140. (In Russian).

    Google Scholar 

  9. Galaktionov VA, Kurdymov P, Samarskii AA. On a parabolic system of quasilinear equations. Part 11 Differ Equat. 1985;21:1544–1559. (In Russian).

    Google Scholar 

  10. Li G, Fan P, Zhu J. Blow-up estimates for a semilinear coupled parabolic system. Appl Math Lett. 2009;22:1297–1302.

    Article  MATH  MathSciNet  Google Scholar 

  11. Lu H, Wang M. Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources. J Math Anal Appl. 2007;333:984–1007.

    Article  MATH  MathSciNet  Google Scholar 

  12. Vazquez JL. An introduction to the mathematical theory of the porous medium equation. In: Delfour MC, Sabidussi G, editors. Shape optimization and free boundaries. Dordrecht: Kluwer Academic; 1992, pp. 347–389.

    Chapter  Google Scholar 

  13. Pan J. The continuous dependence on nonlinearities of solutions of the Neumann problem of a singular parabolic equation. Nonlinear Anal TMA 2007;67:2081–2090.

    Article  MATH  Google Scholar 

  14. Pan J. The expanding behavior of positive set H u (t) of a degenerate parabolic equation. Math Z. 2010;265:817–829.

    Article  MATH  MathSciNet  Google Scholar 

  15. Esteban JR, Rodriguez A, Vazquez JL. A nonlinear heat equation with singular diffusivity. Commun Part Differ Equat. 1988;13:985–1039.

    Article  MATH  MathSciNet  Google Scholar 

  16. Levine HA. The role of critical exponents in blow up theorems. SIAM Rev 1990;32:262–288.

    Article  MATH  MathSciNet  Google Scholar 

  17. Filar M. Boundedness of global solutions for the heat equation with nonlinear boundary condition. Comment Math Univ Carol. 1989;30:479–484.

    Google Scholar 

  18. Mizoguchi N. Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition. J Differ Equat. 2003;193:212–238.

    Article  MATH  MathSciNet  Google Scholar 

  19. Ladyshenskaja OA, Solounikov VA. Uraltseva NN. Linear and quasilinear equations of parabolic type. Translations of mathematical monographs 23. Providence: American Mathematical Society; 1968.

  20. Benilan P, Crandall MG. The continuous dependence on ϕ of solutions of u t − ϕϕ(u) = 0. Indiana Univ Math J. 1981;30:161–177.

    Article  MATH  MathSciNet  Google Scholar 

  21. Admas RA. Sobolev spaces. New York: Academic; 1975.

    Google Scholar 

  22. Varlamov V. Nomlinear heat equation with a fractional Laplacian in disk. Colloq Math. 1999;81:101–122.

    MATH  MathSciNet  Google Scholar 

  23. Deng W, Li Y, Xie C. Global existence and nonexistence for a class of degenerate parabolic systems. Nonlinear Anal, TMA 2003;55:233–244.

    Article  MATH  MathSciNet  Google Scholar 

  24. Ma Z, Wen G. Iterative approximation of solutions for semilinear parabolic equations system. J Math Anal Appl. 2007;209:167–175.

    MATH  MathSciNet  Google Scholar 

  25. Tan Z. Global solution and blowup of semilinear heat equation with critical Sobolev exponent. Comm Partial Differ Equat. 2001;26:717–741.

    Article  MATH  Google Scholar 

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  1. Mathematical Institute, Jimei University, Xiamen, 361021, People’s Republic of China

    Jiaqing Pan

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  1. Jiaqing Pan
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Pan, J. Some Estimates of a System of Degenerate Parabolic Equations. J Dyn Control Syst 20, 1–18 (2014). https://doi.org/10.1007/s10883-013-9205-4

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  • DOI: https://doi.org/10.1007/s10883-013-9205-4

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