Boundary behavior of the fast diffusion equation
HTML articles powered by AMS MathViewer
- by Y. C. Kwong PDF
- Trans. Amer. Math. Soc. 322 (1990), 263-283 Request permission
Abstract:
The fast diffusion equation $\Delta {\upsilon ^m} = {\upsilon _t}$, $0 < m < 1$, is a degenerate nonlinear parabolic equation of which the existence of a unique continuous weak solution has been established. In this paper we are going to obtain a Lipschitz growth rate of the solution at the boundary of $\Omega$ and estimate that in terms of the various data.References
- Philippe Bénilan and Michael G. Crandall, The continuous dependence on $\varphi$ of solutions of $u_{t}-\Delta \varphi (u)=0$, Indiana Univ. Math. J. 30 (1981), no. 2, 161–177. MR 604277, DOI 10.1512/iumj.1981.30.30014
- James G. Berryman and Charles J. Holland, Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal. 74 (1980), no. 4, 379–388. MR 588035, DOI 10.1007/BF00249681 M. Bertsch and L. A. Peletier, Porous media type equations: An overview, Mathematical Institute Publication, no. 7, University of Leiden, 1983.
- Filippo M. Chiarenza and Raul P. Serapioni, A Harnack inequality for degenerate parabolic equations, Comm. Partial Differential Equations 9 (1984), no. 8, 719–749. MR 748366, DOI 10.1080/03605308408820346
- Michael Crandall and Michel Pierre, Regularizing effects for $u_{t}+A\varphi (u)=0$ in $L^{1}$, J. Functional Analysis 45 (1982), no. 2, 194–212. MR 647071, DOI 10.1016/0022-1236(82)90018-0
- Gregorio Díaz and Ildefonso Diaz, Finite extinction time for a class of nonlinear parabolic equations, Comm. Partial Differential Equations 4 (1979), no. 11, 1213–1231. MR 546642, DOI 10.1080/03605307908820126
- Emmanuele DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983), no. 1, 83–118. MR 684758, DOI 10.1512/iumj.1983.32.32008
- Ya Zhe Chen and Emmanuele DiBenedetto, On the local behavior of solutions of singular parabolic equations, Arch. Rational Mech. Anal. 103 (1988), no. 4, 319–345. MR 955531, DOI 10.1007/BF00251444
- M. A. Herrero and J. L. Vázquez, Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math. (5) 3 (1981), no. 2, 113–127 (English, with French summary). MR 646311
- Jerry L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), no. 5, 567–597. MR 477445, DOI 10.1002/cpa.3160280502 N. Krylov and M. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izv. 16 (1981), 151-164.
- Ying C. Kwong, Interior and boundary regularity of solutions to a plasma type equation, Proc. Amer. Math. Soc. 104 (1988), no. 2, 472–478. MR 962815, DOI 10.1090/S0002-9939-1988-0962815-5 —, Asymptotic behaviour of a plasma type equation at finite extinction, Arch. Rational Mech. Anal. 104 (1988).
- Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
- J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727–740. MR 288405, DOI 10.1002/cpa.3160240507 S. Pohozaev, Eigen functions of the equation $\Delta u + \lambda f(u) = 0$, Soviet Math. Dokl. 165 (1965), 1408-1411. E. S. Sabanina, A class of non-linear degenerate parabolic equations, Soviet Math. Dokl. 143 (1962), 495-498.
- Paul E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387–409. MR 696738, DOI 10.1016/0362-546X(83)90092-5 G. Schroeder, Ph. D. Thesis, University of Wisconsin-Madison (to appear).
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 322 (1990), 263-283
- MSC: Primary 35K55; Secondary 35B99
- DOI: https://doi.org/10.1090/S0002-9947-1990-1008697-0
- MathSciNet review: 1008697