Abstract
In this article, the authors establish conditions for the extinction of solutions, in finite time, of the fast diffusion equation \({u_t=\Delta u^m+a\int_\Omega u^p(y,t)\,{d}y,\ 0 < m < 1,}\) in a bounded domain \({\Omega\subset R^N}\) with N > 2. More precisely speaking, it is shown that if p > m, any solution with small initial data vanishes in finite time, and if p < m, the maximal solution is positive in Ω for all t > 0. For the critical case p = m, whether the solutions vanish in finite time or not depends on the value of aμ, where \({\mu=\int_{\Omega}\varphi(x)\,{d}x}\) and \({\varphi}\) is the unique positive solution of the elliptic problem \({-\Delta\varphi(x)=1,\ x\in \Omega; \varphi(x)=0,\ x\in\partial\Omega}\) .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson J.R.: Local existence and uniqueness of solutions of degenerate parabolic equations. Commun. Partial Differential Equations 16, 105–143 (1991)
Anderson J.R., Deng K.: Global existence for degenerate parabolic equations with a nonlocal forcing. Math. Meth. Appl. Sci. 20, 1069–1087 (1997)
Berryman J.G., Holland C.J.: Stability of the separable solution for fast diffusion. Arch. Ration. Mech. Anal. 74, 379–388 (1980)
Borelli M., Ughi M.: The fast diffusion equation with strong absorption: the instantaneous shrinking phenomenon. Rend. Istit. Mat. Univ. Trieste 26, 109–140 (1994)
: Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources. J. Comput. Appl. Math. 202, 237–247 (2007)
Friedman A., Herrero M.A.: Extinction properties of semilinear heat equations with strong absorption. J. Math. Anal. Appl. 124, 530–546 (1987)
Friedman A., Kamin S.: The asymptotic behavior of gas in an N-dimensional porous medium. Trans. Amer. Math. Soc. 262, 551–563 (1980)
Ferreira R., Vazquez J.L.: Extinction behavior for fast diffusion equations with absorption. Nonlinear Anal. 43, 943–985 (2001)
Galaktionov V.A., King J.R.: Fast diffusion equation with critical Sobolev exponent in a ball. Nonlinearity 15, 173–188 (2002)
Galaktionov V.A., Peletier L.A., Vazquez J.L.: Asymptotics of fast-diffusion equation with critical exponent, SIAM J. Math. Anal. 31, 1157–1174 (2000)
Galaktionov V.A., Vazquez J.L.: Asymptotic behaviour of nonlinear parabolic equations with critical exponents A dynamical system approach. J. Funct. Anal. 100, 435–462 (1991)
Galaktionov V.A., Vazquez J.L.: Extinction for a quasilinear heat equation with absorption I. Technique of intersection comparison. Comm. Partial Differential Equations 19, 1075–1106 (1994)
Galaktionov V.A., Vazquez J.L.: Extinction for a quasilinear heat equation with absorption II A dynamical system approach. Comm. Partial Differential Equations 19, 1107–1137 (1994)
Gu Y.G.: Necessary and sufficient conditions of extinction of solution on parabolic equations. Acta. Math. Sinica 37, 73–79 (1994) (in Chinese)
Herrero M.A., Velazquez J.J.L.: Approaching an extinction point in one-dimensional semilinear heat equations with strong absorptions. J. Math. Anal. Appl. 170, 353–381 (1992)
Kalashnikov A.S.: The nature of the propagation of perturbations in problems of non-linear heat conduction with absorption. USSR Comp. Math. Math. Phys. 14, 70–85 (1974)
Kalashnikov A.S.: Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations. Uspekhi Mat. Nauk 42, 135–176 (1987)
Sabinina E.S.: On a class of nonlinear degenerate parabolic equations. Dolk. Akad. Nauk SSSR 143, 794–797 (1962)
Leoni G.: A very singular solution for the porous media equation u t = Δu m−u p when 0 < m < 1. J. Differential Equations 132, 353–376 (1996)
Li Y.X., Wu J.C.: Extinction for fast diffusion equations with nonlinear sources. Electron J. Differential Equations 2005, 1–7 (2005)
Peletier L.A., Junning Z.: Large time behavior of solution of the porous media equation with absorption: The fast diffusion case. Nonlinear Anal. 14, 107–121 (1990)
Peletier L.A., Junning Z.: Source-type solutions of the porous media equation with absorption: The fast diffusion case. Nonlinear Anal. 17, 991–1009 (1991)
Sacks P.E.: Continuity of solutions of a singular parabolic equation. Nonlinear Anal. 7, 387–409 (1983)
Tian Y., Mu C.L.: Extinction and non-extinction for a p-Laplacian equation with nonlinear source. Nonlinear Anal. 69, 2422–2431 (2008)
Yin J.X., Li J., Jin C.H.: Non-extinction and critical exponent for a polytropic filtration equation. Nonl. Anal. 71, 347–357 (2009)
Yin J.X., Jin C.H.: Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources. Proc. Edinburgh Math. Soc. 52, 419–444 (2009)
Yin J.X., Jin C.H.: Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources. Math. Method. Appl. Sci. 30, 1147–1167 (2007)
Yuan H.J. et al.: Extinction and positive for the evolution p-Laplacian equation in R N. Nonl. Anal. TMA 60, 1085–1091 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
The project is supported by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education, by the 985 program of Jilin University and by Graduate Innovation Fund of Jilin University (20111034).
Rights and permissions
About this article
Cite this article
Han, Y., Gao, W. Extinction for a fast diffusion equation with a nonlinear nonlocal source. Arch. Math. 97, 353–363 (2011). https://doi.org/10.1007/s00013-011-0299-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-011-0299-1