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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Existence and nonexistence of global positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary


Authors: Mingxin Wang and Yonghui Wu
Journal: Trans. Amer. Math. Soc. 349 (1997), 955-971
MSC (1991): Primary 35K55, 35K60, 35B35
MathSciNet review: 1401789
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Abstract: This paper deals with the existence and nonexistence of global positive solutions to $u_t=\Delta \ln(1+u)$ in $\Omega \times (0, +\infty )$,

\begin{displaymath}\frac {\partial \ln(1+u)}{\partial n}=\sqrt {1+u}(\ln (1+u))^{\alpha } \quad \text{on}\ \partial \Omega \times (0, +\infty ),\end{displaymath}

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 $\alpha \leq {1/2}$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01864-3
PII: S 0002-9947(97)01864-3
Keywords: Global solutions, blow up, nonlinear diffusion and absorption, upper and lower solutions
Received by editor(s): July 13, 1994
Additional Notes: The first author’s work was supported by The National Natural Science Foundation of China.
Article copyright: © Copyright 1997 American Mathematical Society