Existence and nonexistence of global positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary
HTML articles powered by AMS MathViewer
- by Mingxin Wang and Yonghui Wu PDF
- Trans. Amer. Math. Soc. 349 (1997), 955-971 Request permission
Abstract:
This paper deals with the existence and nonexistence of global positive solutions to $u_t=\Delta \ln (1+u)$ in $\Omega \times (0, +\infty )$, \[ \frac {\partial \ln (1+u)}{\partial n}=\sqrt {1+u}(\ln (1+u))^{\alpha } \quad \text {on} \partial \Omega \times (0, +\infty ),\] and $u(x, 0)=u_0(x)$ in $\Omega$. Here $\alpha \geq 0$ is a parameter, $\Omega \subset \mathbb {R}^N$ is a bounded smooth domain. After pointing out the mistakes in Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, SIAM J. Math. Anal. 24 (1993), 317–326, by N. Wolanski, which claims that, for $\Omega =B_R$ the ball of $\mathbb {R}^N$, the positive solution exists globally if and only if $\alpha \leq 1$, we reconsider the same problem in general bounded domain $\Omega$ and obtain that every positive solution exists globally if and only if $\alpha \leq {1/2}$.References
- Henri Lebesgue, Leçons sur les Constructions Géométriques, Gauthier-Villars, Paris, 1950 (French). MR 0035023
- D. G. Aronson, Regularity properties of flows through porous media: The interface, Arch. Rational Mech. Anal. 37 (1970), 1–10. MR 255996, DOI 10.1007/BF00249496
- D. G. Aronson, Regularity properties of flows through porous media: A counterexample, SIAM J. Appl. Math. 19 (1970), 299–307. MR 265774, DOI 10.1137/0119027
- 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
- Howard A. Levine and Lawrence E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations 16 (1974), 319–334. MR 470481, DOI 10.1016/0022-0396(74)90018-7
- Jeffrey R. Anderson, Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), no. 1, 105–143. MR 1096835, DOI 10.1080/03605309108820753
- Wolfgang Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition, SIAM J. Math. Anal. 6 (1975), 85–90. MR 364868, DOI 10.1137/0506008
- Julián López-Gómez, Viviana Márquez, and Noemí Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Differential Equations 92 (1991), no. 2, 384–401. MR 1120912, DOI 10.1016/0022-0396(91)90056-F
- Ming Xin Wang and Yong Hui Wu, Global existence and blow-up problems for quasilinear parabolic equations with nonlinear boundary conditions, SIAM J. Math. Anal. 24 (1993), no. 6, 1515–1521. MR 1241155, DOI 10.1137/0524085
- Ján Filo, Diffusivity versus absorption through the boundary, J. Differential Equations 99 (1992), no. 2, 281–305. MR 1184057, DOI 10.1016/0022-0396(92)90024-H
- Y.H. Wu, Remarks on the diffusivity versus absorption through the boundary, submitted to J. Sys. Sci. & Math. Scis., in Chinese.
- M.X. Wang, Long time behaviors of solutions of a quasilinear parabolic equation with nonlinear boundary condition, Acta Math. Sinica 39 (1) (1996), 118–124, in Chinese.
- M. Chipot, M. Fila, and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenian. (N.S.) 60 (1991), no. 1, 35–103. MR 1120596
- H. A. Levine, L. E. Payne, P. E. Sacks, and B. Straughan, Analysis of a convective reaction-diffusion equation. II, SIAM J. Math. Anal. 20 (1989), no. 1, 133–147. MR 977493, DOI 10.1137/0520010
- Y.H.Wu, M.X.Wang, Existence and nonexistence of global solution of nonlinear parabolic equation with nonlinear boundary condition, Chinese Ann. of Math.,13(1995), ser. B, 371–378.
Additional Information
- Mingxin Wang
- Affiliation: Department of Mathematics and Mechanics, Southeast University, Nanjing 210018, P.R. China
- Email: mxwang@seu.edu.cn
- Yonghui Wu
- Affiliation: Institute of Applied Physics and Computational Mathematics, Beijing 100088, P.R. China
- Received by editor(s): July 13, 1994
- Additional Notes: The first author’s work was supported by The National Natural Science Foundation of China.
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 955-971
- MSC (1991): Primary 35K55, 35K60, 35B35
- DOI: https://doi.org/10.1090/S0002-9947-97-01864-3
- MathSciNet review: 1401789