Abstract
The aims of this paper are to discuss the extinction and positivity for the solution of the initial boundary value problem and Cauchy problem of u t = div(|∇u m|p−2∇u m). It is proved that the weak solution will be extinct for 1 < p ≤ 1 + 1/m and will be positive for p > 1 + 1/m for large t, where m > 0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Esteban, J. R., Vazquez, J. L.: Homogeneous diffusion in R with power-like nonlinear diffusivity. Arch. Rational Mech. Anal., 103, 39–88 (1988)
Ahmed, N., Sunada, D. K.: Nonlinear flows in porous media. J. Hydraulics. Div. Proc. Amer. Soc. Civil Eng., 95, 1847–1857 (1969)
Kristlanovitch, S. A.: Motion of ground water which does not conform to Darcy’s law. Prikl. Mat. Mech. (Russian), 4, 33–52 (1940)
Volker, R. E.: Nonlinear flow in porous media by finite elements. J. Hydraulics. Div. Proc. Amer. Soc. Civil Eng., 95, 2093–2114 (1969)
Muskat, M.: The flow of homogeneous fluids through porous media, McGraw Hill, New York, 1937
Gilding, B. G., Peletier, L. A.: The Cauchy problem for an equation in the theory of infiltration. Arch. Rational Mech. Anal., 61, 127–140 (1976)
Leibenson, L. S.: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk SSSR, Geography and Geophysics (Russian), 9, 7–10 (1945)
Esteban, J. R., Vazquez, J. L.: On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Anal.T.M.A., 10, 1303–1325 (1986)
Martinson, L. K. Paplov, K. B.: Unsteady shear flows of a conducting fluid with a theological power law. Magnit. Gidrodinamikia, 2, 50–58 (1970)
Antonsev, N.: Axially symmetric problems of gas dynamics with free boundaries. Dokl. Akad. Nauk SSSR, 216, 473–476 (1974)
Ladyzenskaya, O. A.: New equation for the description of incompressible fluids and solvability in the large boundary problem for them. Proc. Steklov Inst. Math., 102, 95–118 (1967)
Liu, Z. H.: On Doubly Degenerate Quasilinear Parabolic Equations of Higher Order. Acta Mathematica Sinica, English Series, 21(1), 197–208 (2005)
Ivanov, A. V.: Hölder estimates for quasilinear doubly degenerate parabolic equations. J. Sov. Math., 56, 2320–2347 (1991)
Manfredi, J. J., Vespri, V.: Large time behavior of solutions to a class of doubly nonlinear parabolic equations. Electronic J. Diff Equa., 02, 1–17 (1994)
Zhao, J. N.: Lipschitz continuity of the free boundary of some nonlinear degenerate parabolic equations. Nonlinear Analysis, 28, 1047–1062 (1997)
Yuan, H. J.: Extinction and positivity for the non-Newtonian polytropic filtration equation. J. Partial Diff. Equations, 9, 169–176 (1996)
Aronson, D. G., Peletier, L. A.: Large time behaviour of solutions of the porous medium equation in bounded domains. J. Diff. Equations, 39, 378–412 (1981)
Evans, L. C.: Application of nonlinear semigroup theory to certain partial differential equations, in “Nonlinear Evolution Equations”(M. G. Crandall, Ed), Academic Press, San Diego, 1978
Lieberman, G. M.: Second order parabolic differential equations, World Scientific, Singapore, New Jersey, London, Hong Kong, 1996
Author information
Authors and Affiliations
Corresponding author
Additional information
The project is supported by NSFC(10371050,10571072) and by the 985 program of Jilin University
Rights and permissions
About this article
Cite this article
Yuan, H.J., Lian, S.Z., Cao, C.L. et al. Extinction and Positivity for a Doubly Nonlinear Degenerate Parabolic Equation. Acta Math Sinica 23, 1751–1756 (2007). https://doi.org/10.1007/s10114-007-0944-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-007-0944-6