The properties of blow-up of solutions to a porous medium equation including nonlinear nonlocal boundary condition

In this paper, we consider the blow-up properties of the positive solutions to a porous medium equation u = Δu^m + c(x, t)u^p for (x, t) ∈ Ω × (0, ∞) with nonlinear nonlocal boundary condition u|∂Ω × (0, ∞) = ∫_Ωf(x, y, t) u^t dy and nonnegative initial data where p > 0 and l > 0. We prove global existence theorem and the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data.