Scalar-field cosmologies with an arbitrary potential

We study the late time evolution of flat and negatively curved FRW models with a perfect fluid matter source and a scalar field having an arbitrary non-negative potential function $V(\phi) .$ We prove using a dynamical systems approach four general results for a large class of non-negative potentials which show that almost always the universe ever expands. In particular, for potentials having a local zero minimum, flat and negatively curved FRW models are ever expanding and the energy density asymptotically approaches zero. We investigate the conditions under which the scalar field asymptotically approaches the minimum of the potential. We discuss the question of whether a closed FRW with ordinary matter can avoid recollapse due to the presence of a scalar field with a non-negative potential.