Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions

This paper deals with the blow-up phenomena for the following porous medium equation systems with nonlinear boundary conditions u t = Δ u m + k 1 ( t ) f 1 ( v ) , v t = Δ v n + k 2 ( t ) f 2 ( u ) i n Ω × ( 0 , t ∗ ) , ∂ u ∂ ν = g 1 ( u ) , ∂ v ∂ ν = g 2 ( v ) o n ∂ Ω × ( 0 , t ∗ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 , v ( x , 0 ) = v 0 ( x ) ≥ 0 i n Ω ¯ , where m , n > 1 , Ω ⊂ R N ( N ≥ 2 ) is bounded convex domain with smooth boundary. Using a differential inequality technique and a Sobolev inequality, we prove that under certain conditions on data, the solution blows up in finite time. We also derive an upper and a lower bound for blow-up time. In addition, as applications of the abstract results obtained in this paper, an example is given.