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

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 {\em Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary}, SIAM J. Math. Anal. {\bf 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 $\alpha\leq {1/2}$. [ABSTRACT FROM AUTHOR]