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: {partial derivative(t)u = Delta u, x is an element of R(+)(N), t > 0, u(x, 0) = phi(x), x is an element of R(+)(N), -partial derivative u/partial derivative x(N) = u(p), x is an element of partial derivative R(+)(N), t > 0, where N >= 1, p > 1 + 1/N, and phi is an element of L(1)(R(+)(N)) boolean AND L(infinity)(R(+)(N)). We prove the existence of global solutions with a small initial data, and study the large time behavior of solutions. (C) 2010 Elsevier Inc. All rights reserved.