In this paper, we concern the fully nonlinear parabolic equations u_t + F(x, t , u, Du, D² u) = 0. Under the natural structure conditions as that in [1], we obtain the C^{1,\alpha} estimates of the viscosity solutions.

This paper concerns the orbital stability of solitary waves of the system of KdV equation coupling with nonlinear Schrödinger equation. By applying the abstract results of Grillakis et al. [1- 2] and detailed spectral analysis, we obtain the stability of the solitary waves.

In this paper we study the scattering theory for the semilincar wave equation u_{tt} - Δu = F(u(t, x), Du(t, x)) in R^n (n ≥ 4) with smooth and small data. We show that the scattering operator exists for the nonlinear term F = F(λ) = O(|λ|^{1, α}), where α is an integer and satisfies α ≥ 2, n = 4; α ≥ I, n ≥ 5.

In this paper, we are concerned with the existence and stability of the positive solutions of a semilinear elliptic system - Δu(x) = a(x)v^6(x) + e(x) - Δv(x) = b(x}u^μ(x) + m(x) \qquad in Ω u = v = 0 \qquad\qquad on ∂Ω where Ω ⊂ R^N is a bounded domain with smooth boundary ∂Ω. It is shown that under the suitable conditions on \delta, μ, there exist a stable and an unstable positive solutions for this system if e and m are sufficiently small in L^∞.

In this paper we consider a class of strongly damped nonlinear wave equations. By the transformation of unknown functions and decomposition of operators, we construct a family of approximate inertial manifolds, and obtain the estimate of orders of approximation of such manifolds to solution orbits.

This paper studies the initial-boundary value problem of GBBM equations  u_t - Δu_t = div f(u) \qquad\qquad\qquad(a) u(x, 0) = u_0(x)\qquad\qquad\qquad(b) u |∂Ω = 0 \qquad\qquad\qquad(c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3, 0 ≤ ϒ < ∞ if n = 2, u_0 (x) ∈ W^{2⋅p}(Ω) ∩ W^{1⋅p}_0(Ω) (2 ≤ p < ∞), then ∀T > 0 there exists a unique global W^{2⋅p} solution u ∈ W^{1,∞}(0, T; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially.

In this paper, we consider the free boundary problem satisfying the Dirichlet boundary condition. This problem is derived from the reaction diffusion equations with the generalized McKean reaction dynamics. We shall show a Hopf bifurcation occurs at some critical point τ when the stationary solution (v^⋅ (x), s^⋅) satisfies 1/3 < s^⋅ < 1.

In this paper, an approach is suggested to consider the relation between the integrable equation and the motion of surface with constant negative Gaussian curvature.