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

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