The first initial-boundary value problem for the following equation u_{tt} - aΔu_{tt} - 2bΔu_t = αΔ^3u - βΔ²u + Δu + ϒΔ(u²) in a unit circle is considered. The existence of strong solution is established in the space C^0([0, ∞), H^s_r (0, 1)), s < 7/2, and the solutions are constructed in the form of series in the small parameter present in the initial conditions. For 5/2 < s < 7/2, the uniqueness is proved. The long-time asymptotics is obtained in the explicit form.

In this paper, we prove that there exists a unique global smooth solution for the homogeneous Neumann boundary value problem of the Landau-Lifschitz equation if the initial function is smooth.

In this paper, the singular semilinear elliptic equation Δu + q(x)u^α + p(x)u^{-β} - h(x)u^{-ϒ} = 0, x ∈ R^N, N ≥ 3, is studied via the super and sub-solution method, where Δ is the Laplacian operator, α ∈ [0, 1), β > 0, and ϒ ≥ 1 are constants. Under a set of suitable assumptions on functions q(x), p(x) and h(x), it is proved that there exists for the equation one and only one minimal positive entire solution.

This paper is concerned with a strongly coupled prey-predator model with homogeneous Dirichlet boundary conditions. The existence and uniqueness of coexistence states are discussed.

We study the Dirichlet initial-boundary value problem of the general- ized Kuramoto-Sivashinsky equation u_t + u_{xxxx} + λu_{xx} + f(u)_ x = 0 on the interval [0, l]. The nonlinear function f satisfies the condition |f'(u)| ≤ c|u|^{α-1} for some α > 1. We prove that if λ < \frac{4π²}{l²}, then the strong solution is global and exponentially decays to zero for any initial datum u_0 ∈ H²_0 (0, l) if 1 ‹ α ≤ 7, and for small u_0 ∈ H²_0 (0, l) if α › 7. We then consider the equation u_t + u_{xxxx} + λu_{xx} + μu + au_{xxx} + bu_x = F(u, u_x, u_{xx}, u_{xxx}). We prove that if F is twice differentiable, ∇²F is Lipschitz continuous, and F(0) = ∇F(0) = 0, and if λ and μ satisfy μ + σ(λ) > 0 (σ(λ)=the first eigenvalue of the operator \frac{d^4}{dx^4} + λ\frac{d²}{dx²}), then the solution for small initial datum is global and exponentially decays to zero.

In this paper we study the existence of global solutions to the Cauchy problem of nonlinear Schrödinger equation by establishing time weight function spaces and using the contraction mapping principle.