  This paper studies the large time existence for the motion of closed
hypersurfaces in a radially symmetric potential. In physical, this surface can
be considered as an electrically charged membrane with a constant charge per
area in a radially symmetric potential. The evolution of such surface has been
investigated by Schn\"urer and Smoczyk (Evolution of hypersurfaces in central
force fields, J. Reine Angew. Math. 550 (2002), 77-95). To study its motion, we
introduce a quasi-linear degenerate hyperbolic equation which describes the
motion of the surfaces extrinsically. Our main results show that the large time
existence of such Cauchy problem and the stability with respect to small
initial data. When the radially symmetric potential function $v\equiv1$, the
local existence and stability results have been obtained by Notz (Closed
Hypersurfaces driven by mean curvature and inner pressure, Comm. Pure Appl.
Math. 66(5) (2013), 790-819). The proof is based on a new Nash-Moser iteration
scheme.
