The geometric properties of a degenerate parabolic equation with periodic source term

In this paper, we discuss the geometric properties of solution and lower bound estimate of ∆um^−1 of the Cauchy problem for a degenerate parabolic equation with periodic source term ut =∆u^m+ u^psin t. Our objective is to show that: (1)with continuous variation of time t, the surface ϕ = [u(x,t)]mδ/q is a complete Riemannian manifold floating in space R^N+1and is tangent to the space R^N at ∂H0(t); (2)the surface u = u(x,t) is tangent to the hyperplane W(t) at ∂Hu(t).