Global existence of solutions for the heat equation with a nonlinear boundary condition

We consider the initial–boundary value problem for the heat equation with a nonlinear boundary condition: { ∂ t u = Δ u , x ∈ R + N , t > 0 , u ( x , 0 ) = φ ( x ) , x ∈ R + N , − ∂ u ∂ x N = u p , x ∈ ∂ R + N , t > 0 , where N ⩾ 1 , p > 1 + 1 / N , and φ ∈ L 1 ( R + N ) ∩ L ∞ ( R + N ) . We prove the existence of global solutions with a small initial data, and study the large time behavior of solutions.