DOI QR코드

DOI QR Code

EXTINCTION AND NON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSIVE p-LAPLACE EQUATION WITH A NONLOCAL SOURCE

  • Han, Yuzhu (Institute of Mathematics Jilin University) ;
  • Gao, Wenjie (Department of Mathematics Jilin University) ;
  • Li, Haixia (Department of Mathematics Jilin University)
  • Received : 2012.05.16
  • Published : 2014.01.31

Abstract

In this paper, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive p-Laplace equation $u_t=div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+a{\int}_{\Omega}u^q(y,t)dy$, 1 < p < 2, in a bounded domain ${\Omega}{\subset}R^N$ with $N{\geq}1$. More precisely, it is shown that if q > p-1, any solution vanishes in finite time when the initial datum or the coefficient a or the Lebesgue measure of the domain is small, and if 0 < q < p-1, there exists a solution which is positive in ${\Omega}$ for all t > 0. For the critical case q = p-1, whether the solutions vanish in finite time or not depends crucially on the value of $a{\mu}$, where ${\mu}{\int}_{\Omega}{\phi}^{p-1}(x)dx$ and ${\phi}$ is the unique positive solution of the elliptic problem -div(${\mid}{\nabla}{\phi}{\mid}^{p-2}{\nabla}{\phi}$) = 1, $x{\in}{\Omega}$; ${\phi}(x)$=0, $x{\in}{\partial}{\Omega}$. This is a main difference between equations with local and nonlocal sources.

References

  1. J. G. Berryman and C. J. Holland, Stability of the separable solution for fast diffusion, Arch. Ration. Mech. Anal. 74 (1980), no. 4, 379-388.
  2. E. Dibenedetto, Degenerate Parabolic Equations, Springer, New York, 1993.
  3. R. Ferreira and J. L. Vazquez, Extinction behavior for fast diffusion equations with absorption, Nonlinear Anal. 43 (2001), 943-985. https://doi.org/10.1016/S0362-546X(99)00178-9
  4. A. Friedman and Miguel A. Herrero, Extinction properties of semilinear heat equations with strong absorption, J. Math. Anal. Appl. 124 (1987), no. 2, 530-546. https://doi.org/10.1016/0022-247X(87)90013-8
  5. A. Friedman and S. Kamin, The asymptotic behavior of gas in an n-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980), no. 2, 551-563.
  6. J. Furter and M. Crinfeld, Local vs. nonlocal interactions in population dynamics, J. Math. Biol. 27 (1989), no. 1, 65-80. https://doi.org/10.1007/BF00276081
  7. V. A. Galaktionov and J. R. King, Fast diffusion equation with critical Sobolev exponent in a ball, Nonlinearity 15 (2002), no. 1, 173-188. https://doi.org/10.1088/0951-7715/15/1/308
  8. V. A. Galaktionov, L. A. Peletier, and J. L. Vazquez, Asymptotics of the fast-diffusion equation with critical exponent, SIAM J. Math. Anal. 31 (2000), no. 5, 1157-1174. https://doi.org/10.1137/S0036141097328452
  9. V. A. Galaktionov and J. L. Vazquez, Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach, J. Funct. Anal. 100 (1991), no. 2, 435-462. https://doi.org/10.1016/0022-1236(91)90120-T
  10. V. A. Galaktionov and J. L. Vazquez, Extinction for a quasilinear heat equation with absorption I. Technique of intersection comparison, Comm. Partial Differential Equations 19 (1994), no. 7-8, 1075-1106. https://doi.org/10.1080/03605309408821046
  11. V. A. Galaktionov and J. L. Vazquez, Extinction for a quasilinear heat equation with absorption II. A dynamical system approach, Comm. Partial Differential Equations 19 (1994), no. 7-8, 1107-1137. https://doi.org/10.1080/03605309408821047
  12. Y. G. Gu, Necessary and sufficient conditions of extinction of solution on parabolic equations, Acta. Math. Sinica 37 (1994), 73-79 (in Chinese).
  13. Y. Z. Han and W. J. Gao, Extinction for a fast diffusion equation with a nonlinear nonlocal source, Arch. Math. (Basel) 97 (2011), no. 4, 353-363. https://doi.org/10.1007/s00013-011-0299-1
  14. M. A. Herrero and J. J. L. Velazquez, Approaching an extinction point in one-dimensional semilinear heat equations with strong absorptions, J. Math. Anal. Appl. 170 (1992), no. 2, 353-381. https://doi.org/10.1016/0022-247X(92)90024-8
  15. C. H. Jin, J. X. Yin, and Y. Y. Ke, Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources, Proc. Edinb. Math. Soc. (2) 52 (2009), no. 2, 419-444. https://doi.org/10.1017/S0013091507000399
  16. A. S. Kalashnikov, The nature of the propagation of perturbations in problems of nonlinear heat conduction with absorption, USSR Comp. Math. Math. Phys. 14 (1974), 70-85. https://doi.org/10.1016/0041-5553(74)90073-1
  17. A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk 42 (1987), no. 2, 135-176.
  18. F. C. Li and C. H. Xie, Global and blow-up solutions to a p-Laplacian equation with nonlocal source, Comput. Math. Appl. 46 (2003), no. 10-11, 1525-1533. https://doi.org/10.1016/S0898-1221(03)90188-X
  19. Y. X. Li and J. C. Wu, Extinction for fast diffusion equations with nonlinear sources, Electron J. Differential Equations 2005 (2005), no. 23, 1-7.
  20. W. Liu, Extinction and non-extinction of solutions for a nonlocal reaction-diffusion problem, Electron. J. Qual. Theory Differ. Equ. 2010 (2010), no. 15, 1-12.
  21. W. Liu and B. Wu, A note on extinction for fast diffusive p-Laplacian with source, Math. Methods Appl. Sci. 31 (2008), no. 12, 1383-1386. https://doi.org/10.1002/mma.976
  22. E. S. Sabinina, On a class of non-linear degenerate parabolic equations, Dolk. Akad. Nauk SSSR 143 (1962), 794-797.
  23. Y. Tian and C. L. Mu, Extinction and non-extinction for a p-Laplacian equation with nonlinear source, Nonlinear Anal. 69 (2008), no. 8, 2422-2431. https://doi.org/10.1016/j.na.2007.08.021
  24. J. X. Yin and C. H. Jin, Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources, Math. Methods Appl. Sci. 30 (2007), no. 10, 1147-1167. https://doi.org/10.1002/mma.833
  25. J. X. Yin, J. Li, and C. H. Jin, Non-extinction and critical exponent for a polytropic filtration equation, Nonlinear Anal. 71 (2009), no. 1-2, 347-357. https://doi.org/10.1016/j.na.2008.10.082
  26. H. J. Yuan, S. Z. Lian, W. J. Gao, X. J. Xu, and C. L. Cao, Extinction and positivity for the evolution p-Laplacian equation in $R^N$, Nonlinear Anal. 60 (2005), no. 6, 1085-1091. https://doi.org/10.1016/j.na.2004.10.009

Cited by

  1. A complete characterization of nonlinear absorption for the evolution p-Laplacian equations to have positive or extinctive solutions vol.71, pp.8, 2016, https://doi.org/10.1016/j.camwa.2016.02.021
  2. Critical extinction exponent for a quasilinear parabolic equation with a gradient source vol.48, pp.1-2, 2015, https://doi.org/10.1007/s12190-014-0805-2