Global Existence, Extinction, and Non-Extinction of Solutions to a Fast Diffusion p-Laplace Evolution Equation with Singular Potential

Abstract

In this paper, we study a class of fast diffusion p-Laplace equation with singular potential in a bounded smooth domain with homogeneous Dirichlet boundary condition. By using energy estimates, Hardy-Littlewood-Sobolev inequality, and some ordinary differential inequalities, we get the solution of the equation exists globally. Moreover, the conditions of extinction and non-extinction are studied. The results of this paper extend and complete the previous studies on this equation.

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References

  1. 1.

    Arrieta JM, Rodriguez-Bernal A, Souplet P. Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. Ann Sc Norm Super Pisa Cl Sci (5) 2004;3(1):1–15.

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Attouchi A. Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion. J Differential Equations 2012;253 (8):2474–2492.

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Badiale M, Tarantello G. A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch Ration Mech Anal 2002;163(4): 259–293.

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Ben-Artzi M, Souplet P, Weissler FB. The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces. J Math Pures Appl (9) 2002;81(4):343–378.

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    DiBenedetto E. Degenerate parabolic equations. Universitext. New York: Springer-Verlag; 1993.

    Google Scholar 

  6. 6.

    Evans LC, Knerr BF. Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities. Illinois J Math 1979;23(1):153–166.

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Friedman A, Herrero MA. Extinction properties of semilinear heat equations with strong absorption. J Math Anal Appl 1987;124(2):530–546.

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Gu YG. Necessary and sufficient conditions for extinction of solutions to parabolic equations. Acta Math Sinica 1994;37(1):73–79.

    MathSciNet  Google Scholar 

  9. 9.

    Guo B, Gao W. Non-extinction of solutions to a fast diffusive p-Laplace equation with Neumann boundary conditions. J Math Anal Appl 2015;422(2):1527–1531.

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Guo J-S, Hu B. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete Contin Dyn Syst 2008;20(4):927–937.

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Herrero MA, Velázquez JJL. Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption. J Math Anal Appl 1992;170(2):353–381.

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Hesaaraki M, Moameni A. Blow-up positive solutions for a family of nonlinear parabolic equations in general domain in \(\mathbb {R}^{N}\). Michigan Math J 2004;52(2):375–389.

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Kalašnikov AS. The nature of the propagation of perturbations in problems of nonlinear heat conduction with absorption. ž Vyčisl Mat i Mat Fiz 1974;1075: 14:891–905.

    MathSciNet  Google Scholar 

  14. 14.

    Lair AV. Finite extinction time for solutions of nonlinear parabolic equations. Nonlinear Anal 1993;21(1):1–8.

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Liao M, Gao W. Blow-up phenomena for a nonlocal p-Laplace equation with Neumann boundary conditions. Arch Math (Basel) 2017;108(3):313–324.

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Liu W, Wu B. A note on extinction for fast diffusive p-Laplacian with sources. Math Methods Appl Sci 2008;31(12):1383–1386.

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Quittner P, Souplet P. 2007. Superlinear parabolic problem. Birkhäuser advanced texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel. Blow-up, global existence and steady states.

  18. 18.

    Sattinger D H. On global solution of nonlinear hyperbolic equations. Arch Rational Mech Anal 1968;30:148–172.

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Souplet P, Vázquez JL. Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem. Discrete Contin Dyn Syst 2006;14(1):221–234.

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Tan Z. Non-Newton filtration equation with special medium void. Acta Math Sci Ser B (Engl Ed) 2004;24(1):118–128.

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Ya T, Mu C. Extinction and non-extinction for a p-Laplacian equation with nonlinear source. Nonlinear Anal 2008;69(8):2422–2431.

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Vázquez JL. 2007. The porous medium equation. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford. Mathematical theory.

  23. 23.

    Wang Y. The existence of global solution and the blowup problem for some p-Laplace heat equations. Acta Math Sci Ser B (Engl Ed) 2007;27(2):274–282.

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Wu Z, Zhao J, Yin J, Li H. 2001. Nonlinear diffusion equations. World Scientific Publishing Co., Inc., River Edge, NJ. Translated from the 1996 Chinese original and revised by the authors.

  25. 25.

    Xu B, Yuan R. The existence of positive almost periodic type solutions for some neutral nonlinear integral equation. Nonlinear Anal 2005;60(4):669–684.

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Zhan H. Hölder inequality applied on a non-Newtonian fluid equation with a nonlinear convection term and a source term. J Inequal Appl 2018;344:21.

    Google Scholar 

  27. 27.

    Zhan H. The uniqueness of a nonlinear diffusion equation related to the p-Laplacian. J Inequal Appl 2018;7:14.

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Zhan H, Feng Z. Stability of the solutions of a convection-diffusion equation. Nonlinear Anal 2019;182:193–208.

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Zhao JN. Existence and nonexistence of solutions for ut = div(|∇u|p− 2u) + f(∇u, u, x, t). J Math Anal Appl 1993;172(1):130–146.

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Zhou J. A multi-dimension blow-up problem to a porous medium diffusion equation with special medium void. Appl Math Lett 2014;30:6–11.

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Zhou J. Global existence and blow-up of solutions for a non-Newton polytropic filtration system with special volumetric moisture content. Comput Math Global Appl 2016;71(5):1163–1172.

    MathSciNet  Article  Google Scholar 

  32. 32.

    Zhu L. Complete quenching phenomenon for a parabolic p-Laplacian equation with a weighted absorption. J Inequal Appl 2018;248:16.

    MathSciNet  Google Scholar 

  33. 33.

    Zhu L. The quenching behavior of a quasilinear parabolic equation with double singular sources. C R Math Acad Sci Paris 2018;356(7):725–731.

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Jun Zhou.

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This work is supported by NSFC (Grant No. 11201380).

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Deng, X., Zhou, J. Global Existence, Extinction, and Non-Extinction of Solutions to a Fast Diffusion p-Laplace Evolution Equation with Singular Potential. J Dyn Control Syst 26, 509–523 (2020). https://doi.org/10.1007/s10883-019-09462-5

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Keywords

  • p-Laplace equation
  • Global existence
  • Extinction
  • Non-extinction

Mathematics Subject Classification (2010)

  • 35B40
  • 35K35
  • 35K58