Applying the homogeneous balance method, we have found the explicit and soliton solutions and given a successive formula of finding explicit solutions to the (2+1)-dimensional Broer-Kaup equations. Moreover, by using the Lie group method, we have discussed the similarity solutions to the (2+1)-dimensional Broer-Kaup equations.

In this paper, the authors consider complex Ginzburg-Landau equation (CGL) in three spatial dimensions u_t = ρu + (1 + iϒ)Δu - (1 + iμ) |u|^{2σ u + f, where u is an unknown complex-value function defined in 3+1 dimensional space-time R^{3+1}, Δ is a Laplacian in R^3, Δ > 0, ϒ, μ are real parameters, Ω ∈ R^3 is a bounded domain. By using the method of Galërkin and Faedo-Schauder fix point theorem we prove the existence of approximate solution uN of the problem. By establishing the uniform boundedness of the norm ||uN|| and the standard compactness arguments, the convergence of the approximate solutions is considered. Moreover, the existence of the periodic solution is obtained .

The existence and uniqueness of the positive solution for the generalized Lotka-Volterra competition model for several competing species Δu^i + u^i(a - g(u¹, ..., uN^)) = 0 in Ω , u^i = 0 on ∂Ω , for i = 1, ...,N were investigated. The techniques used in this paper are elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.

In this note, we establish some estimates of solutions of the scalar Ginzburg-Landau equation and other nonlinear Laplacian equation Δu = f(x, u). This will give an estimate of the Hausdorff dimension for the free boundary of the obstacle problem.

In this paper, we study the asymptotic behavior of the solutions to the bipolar hydrodynamic model with Dirichlet boundary conditions. It is shown that the initial boundary problem of the model admits a global smooth solution which decays to the steady state exponentially fast.

In this paper, we study the non-negative solutions to a degenerate parabolic system with nonlinear boundary conditions in the multi-dimensional case. By the upper and lower solutions method, we give the conditions on the existence and non-existence of global solutions.

In this paper, we study the existence of the global attractor A" of reaction-diffusion equation ∂_tu^ε(x, t) = A_εu^ε(x, t) - f(x, ε^{-1}x, u^ε(x, t)), and the homogenized attractor A^0 of the corresponding homogenized equation, then give explicit estimates for the distance between the attractor A^ε and the homogenized attractor A^0 .

We discuss the solution of the Cauchy problem of the generalized Davey-Stewartson equation. When the initial value is small enough, we obtain the global wellposedness of the solution and scattering.