Journal of the
Korean Mathematical Society JKMS

In this paper, we study the properties of positive solutions for the reaction-diffusion equation $u_t =\Delta u+\int_\Omega {u^p} dx-ku^q$ in $ \Omega \times \left( {0,T } \right)$ with nonlocal nonlinear boundary condition $u\left( {x,t} \right)=\int_\Omega {f\left( {x,y} \right)u^l\left( {y,t} \right)} dy$ on $\partial \Omega \times \left( {0,T } \right)$ and nonnegative initial data $u_0 \left( x \right)$, where $p$, $q$, $k$, $l>0$. Some conditions for the existence and nonexistence of global positive solutions are given.

Keywords: reaction-diffusion equation, global existence, blow-up, nonlocal nonlinear boundary condition

MSC numbers: 35B35, 35K57, 35K60, 35K65