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:{∂tu=Δu,x∈R+N,t>0,u(x,0)=φ(x),x∈R+N,−∂u∂xN=up,x∈∂R+N,t>0, where N⩾1N⩾1, p>1+1/Np>1+1/N, and φ∈L1(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.