The paper deal with the existence of positive solution for the following (p,q)-Laplacian nonlinear system \begin{align*} \left\{  \begin{array}{ll} -Δ_pu=a(x)(α_1f(v)+β_1h(u)), & x∈Ω,\\ -Δ_qv=b(x)(α_2g(u)+β_2k(v)),& x∈Ω,\\ u=v=0,& x∈∂Ω, \end{array} \right. \end{align*} where $Δ_p$ denotes the p-Laplacian operator defined by $Δ_{p}z=div(|∇_z|^{p-2}∇z), p>1, α_1, α_2, β_1, β_2$ are positive parameters and Ω is a bounded domain in $R^N(N > 1)$ with smooth boundary ∂Ω. Here a(x) and b(x) are $C^1$ sign-changing functions that maybe negative near the boundary and f, g, h, k are C^1 nondecreasing functions such that $f, g, h, k: [0,∞)→[0,∞); f (s), g(s), h(s), k(s) > 0; s > 0$ and $lim_{n→∞}\frac{f(Mg(x)^{\frac{1}{q-1}}}{x^{p-1}}=0$ for every $M > 0$.  We discuss the existence of positive solution when $f, g, h, k, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.

A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is also decoupled in computation, whichmeans that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in implementation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L^∞- norm, and for N in the discrete L^2-norm, respectively, where U and N are the numerical solutions of the KGZ equation. Numerical results verify the theoretical analysis.

In this paper we present some results concerning the optimal shape design problem governed by the fourth-order variational inequalities. The problem can be considered as a model example for the design of the shapes for elastic-plastic problem. The computations are done by finite element method, and the performance criterion is minimized by the material derivative method. We also discuss the error estimates in the appropriate norm and present some numerical results. An example is used to clearly illustrate the essential elements of shape design problems.

This paper is devoted to studying the following initial-boundary value problemfor one-dimensional semilinearwave equationswith variable coefficients andwith subcritical exponent: $u_{tt}-∂_x(a(x)∂_xu)=|u|^p, x > 0, t > 0, n=1,$ where $u=u(x,t)$ is a real-valued scalar unknown function in $[0,+∞)×[0,+∞)$, here a(x) is a smooth real-valued function of the variable $x∈(0,+∞)$. The exponents p satisfies $1 < p < +∞$ in (0.1). It is well-known that the number $p_c(1)=+∞$ is the critical exponent of the semilinear wave equation (0.1) in one space dimension (see for e.g., [1]). We will establish a blowup result for the above initial-boundary value problem, it is proved that there can be no global solutions no matter how small the initial data are, and also we give the lifespan estimate of solutions for above problem.

We present an approximation method for the non-stationary nonlinear incompressible Navier-Stokes equations in a cylindrical domain (0,T)×G,where G⊂R^3 is a smoothly bounded domain. Ourmethod is applicable to general three-dimensional flow without any symmetry restrictions and relies on existence, uniqueness and representation results from mathematical fluid dynamics. After a suitable time delay in the nonlinear convective term v·∇v we obtain globally (in time) uniquely solvable equations, which - by using semi-implicit time differences - can be transformed into a finite number of Stokes-type boundary value problems. For the latter a boundary element method based on a corresponding hydrodynamical potential theory is carried out. The method is reported in short outlines ranging from approximation theory up to numerical test calculations.

Let D⊂R^N(N ≥ 3), be a smooth bounded domain with smooth boundary ∂D. In this paper, the existence of boundary blow-upweak solutions for the quasilinear elliptic equation Δ_pu=λk(x) f (u) in D(λ > 0 and 1 < p < N), is obtained under new conditions on k. We give also asymptotic behavior near the boundary of such solutions.