In this paper, the Cauchy problems for a class of nonlinear degenerate hyperbolic Monge-Ampere equations are studied and some results on local isometric embedding in R^3 of two dimensional Riemannian manifolds with non positive curvature, are contained.

Consider the degenerate parabolic equation (porous medium equation with gravity term): u_t = (u^m)_{xx} + (u^n)_x, -∞ < x < ∞, t > 0, m > 1 u(x, 0) = u_0(x), -∞ < x < ∞ The main results consist of the estimation of t^∗_i called waiting time, the behavior of pressure V = \frac{m}{m-1}u^{m-1} near a vertical or a nonvertical part of ς_i(t) and a condition of that ς_i(t) is continuously differentiable.

In this paper we consider the elliptic equation Δu + K(x)e^{2u} = f(x), which arises from prescribed curvature problem in Riemannian geometry. It is proved that if K(x) is negative and continuous in R², then for any f ∈ L²_{loc} (R²) such that f(x) ≤ K(x), the equation possesses a positive solution. A uniqueness theorem is also given.

In this paper, we extend the results in [4] to general p(v), and remove the condition (A) for existence, but keep it for nonexistence.

In this paper the extinction properties of solutions to both linear and nonlinear hyperbolic equations, which reflect the effect of damping boundary conditions and have not been explicitly stated in literature before, are introduced and extensively studied.

It is proved that when a progressing wave described by conormal distribution for a quasilinear hyperbolic 2 × 2 system hits a solid wall transvenally, the reflected wave remains conormal. In contrast to the semilinear case, such conormal distributions had to be defined inductively to take into account of the fact that the relevant characteristic surfaces are not necessarily smooth. The argument involves a suitable coordinate change to reduce the problem to a simple form and an iterative induction on the tangential regularity of the solution as well as that of the characteristic surfaces at which the wave fronts are situated.

In this paper, we consider the one-dimensional quasilinear Verigin problem and the relative diffraction problem. The existence, uniqueness, and the smoothness of the global solution are discussed.