A one-dimensional semilinear parabolic equation with a nonlinear boundary condition is studied. Some results on steady states, global existence and blow-up are established. A more complete study of a slightly more general problem (also in higher dimension) was carried out independently in M. Chipot, M. Fila and P. Quittner [Acta Math. Univ. Comen., New Ser. 60, No. 1, 35-103 (1991; Zbl 0743.35038)] and continued in P. Quittner [Comment. Math. Univ. Carol. 34, No. 1, 105-124 (1993; Zbl 0794.35089)].