On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

In this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary ∂Ω and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1).     

On the Neumann problem with the Hardy–Sobolev potential

In this paper we investigate the solvability of the nonlinear Neumann problem (1.1) with indefinite weight functions and a critical Hardy–Sobolev nonlinearity. We examine the common effect of the shape of the graph of a weight function and the mean curvature of the boundary on the existence of solutions of problem (1.1). We also investigate the regularity of solutions.      

On the ZK equation with a directional dissipation

This paper studies the generalized Zakharov-Kuznetsov-Burgers equation. The initial value problem associated to this equation will be investigated in the nonhomogeneous Sobolev spaces and some suitable weighted spaces, under appropriate conditions. Moreover, an ill-posedness result (in some sense) will be proved in the anisotropic Sobolev spaces. Furthermore some exact traveling wave solutions of this equation will be obtained.     

On an obstacle problem for degenerate elliptic operators involving the critical Sobolev exponent

We establish the existence of a solution to the variational inequality (the obstacle problem) (1.1) which involves the critical Sobolev exponent. This result is also extended to an obstacle problem with a lower order perturbation. Dedicated to Professor F. Browder on the occasion of his 80-th birthday     

Subcritical Hamilton–Jacobi fractional equation in

Solvability of Cauchy's problem in for fractional Hamilton–Jacobi equation (1.1) with subcritical nonlinearity is studied here both in the classical Sobolev spaces and in the locally uniform spaces. The first part of the paper is devoted to the global in time solvability of subcritical equation (1.1) in locally uniform phase space, a generalization of the standard Sobolev spaces. Subcritical growth of the nonlinear term with respect to the gradient is considered. We prove next the global in time solvability in classical Sobolev spaces, in Hilbert case. Regularization effect is used there to guarantee global in time extendibility of the local solution. Copyright © 2014 John Wiley & Sons, Ltd.     

RN上一类半线性椭圆方程组的多重解

利用对称山路引理,L2(RN)上加权Sobolev空间紧嵌入定理和一个线性耦合特征值问题,本文证明了RN上一类半线性椭圆方程组的多重解的存在性.     

R~2中具有趋化性的抛物-椭圆系统的解的性态分析

在二维空间中讨论了一个抛物-椭圆系统,而该系统来源于生物学中的趋化性模型.主要在Sobolev空间的框架下讨论了解的全局存在性与解的爆破性质,得出结论该系统存在一个门槛值,而该值决定了解全局存在或者发生爆破.最后利用利李亚普诺夫函数给出了定理的证明并得出结论.     

A smoothing effect and polynomial growth of the Sobolev norms for the KP-II equation

We prove that the eventual growth in time of the Sobolev norms of the solutions of the KP-II equation is at most polynomial.     

Condensation of Least-energy Solutions of a Semilinear Neumann Problem

This paper is devoted to the study of tho least-energy solutions of a singularly perturbed Neumann problem involving critical Sobolev exponents. The condensation rate is given when n > 4 apd an asymptotic behavior result is obtained.     

On a Class of Schrödinger-Type Equations with Indefinite Weight Functions

In this paper we investigate the solvability of the Schrödinger equation (1.1) with indefinite weight functions and a subcritical Sobolev nonlinearity. We examine the common effect of the properties of the Nehari manifold and the fibrering maps on the existence of solutions of problem (1.1).     

具有趋化性的非线性系统的解的爆破性态分析

讨论了一个非线性的抛物-椭圆系统,而该系统来源于生物数学中的一个趋化性模型.主要在Sobolev空间的框架下讨论了系统解的爆破性质,得出结论在二维空间中该系统存在一个门槛值,而该值决定了解全局存在或者是发生爆破.最后利用利亚普诺夫函数、下解爆破等方法给出了定理的证明并得出结论.     

The Cauchy Problem for the Kadomtsev-Petviashvili (KPII) Equation in Three Space Dimensions

We prove the existence and uniqueness of local solutions in time for the Kadomtsev-Petviashvili equation (KPII) in ?³ for initial data in a suitable anisotropic Sobolev space of functions which have essentially 1⁺ derivative in the variable x and 0⁺ derivatives in the variables y and z.     

一类弱耗散Camassa-Holm方程局部强解的适定性及弱解的存在性

使用Pseudoparabolic正则化方法和从弱耗散Camassa-Holm方程自身导出的估计式,在Sobolev空间Hs(R)(s3/2)中,证明了该Camassa-Holm方程解的局部适定性.同时给出了一个在空间Hs(R)(1s2\3)中确保该方程弱解存在的充分条件.     

Global well-posedness and existence of uniform attractor for magnetohydrodynamic equations

We study the global well-posedness and existence of uniform attractor for magnetohydrodynamic (MHD) equations. The hydrodynamic system consists of the Navier–Stokes equations for the fluid velocity and pressure coupled with a reduced from of the Maxwell equations for the magnetic field. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the magnetic field is subject to a time-dependent Dirichlet boundary condition. We first establish the global existence of weak and strong solutions to Equations (1.1)-(1.4). And at this stage, we further derive the existence of a uniform attractor for Equations (1.1)-(1.4).     

The critical Neumann problem for semilinear elliptic equations with concave perturbations

We consider the semilinear Neumann problem involving the critical Sobolev exponent with an indefinite weight function and a concave purturbation. We prove the existence of two distinct solutions.      

Non-Uniform Dependence for the Periodic CH Equation

We show that the solution map of the periodic CH equation is not uniformly continuous in Sobolev spaces with exponent greater than 3/2. This extends earlier results to the whole range of Sobolev exponents for which local well-posedness of CH is known. The crucial technical tools used in the proof of this result are a sharp commutator estimate and a multiplier estimate in Sobolev spaces of negative index.     

Solutions for a class of singular nonlinear boundary value problem involving critical exponent

In this paper, we consider the existence of multiple solutions for a class of singular nonlinear boundary value problem involving critical exponent in Weighted Sobolev Spaces. The existence of two solutions is established by using the Ekeland Variational Principle. Meanwhile, the uniqueness of positive solution for the same problem is also obtained under different assumptions.     

Forward–backward SDEs with distributional coefficients

Forward–backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of solution and show its existence and in certain cases its uniqueness. Moreover we establish a link with PDE theory via a non-linear Feynman–Kac formula. The associated semi-linear parabolic PDE is the same for both FBSDEs, also involves distributional coefficients and has not previously been investigated.     

A new class of nonhomogeneous differential operator and applications to anisotropic systems

We introduce a new class of operators that extend both generalized Laplace operators and generalized mean curvature operators. We start the discussion on general anisotropic systems with variable exponents that involve our operators, then we focus on a specific example of such system, we show that it admits a unique weak solution and we complete our work with some comments on other related systems. The newly introduced operators are appropriate for the study conducted in the anisotropic spaces with variable exponents, but at the end of the paper we also provide their versions corresponding to the studies conducted in the anisotropic Sobolev spaces with constant exponents, or in the isotropic variable exponent Sobolev spaces, since, to the best of our knowledge, they represent a novelty even for the classical Sobolev spaces.