Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer Π _T =ℝ ⁿ ×[0,T] for the equationu _t =cGΔu _t +Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC ¹(ℝ₊) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈C _x,t ^2,1 (Π _T ) with the properties: , . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996. This research was partially supported by the International Science Foundation under grant No. MX6000.     

A fundamental solution to the Cauchy problem for a fourth-order pseudoparabolic equation

The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.     

一类一维非线性伪抛物型方程的反问题

将Riemann函数方法与不动点理论有效地结合起来,研究了一类一维非线性伪抛物型方程的后向热流问题,得出了反问题解的存在唯一性结论.     

耦合Schrdinger-KdV方程组Cauchy问题的适定性

本文研究了耦合Schrodinger－KdV方程组的Cauchy问题,此耦合方程组刻化了一维Langmuir和离子声波相互作用的非线性动力学行为．本文建立了此问题在Hk×Hk中的整体适定性理论（k∈Z+）．     

一般耦合Maxwell－SchrOdinger方程的Cauchy问题（1）

本文研究了一般耦合Ｍａｘｗｅｌｌ－Ｓｃｈｒｏｄｉｎｇｅｒ方程的Ｃａｕｃｈｙ问题,采用正则化方法,借助于推广的Ｈ．Ｐｅｃｈｅｒ引理及其它分析工具,在光滑解的框架下,建立了任意维空间中一般耦合Ｍａｘｗｅｌｌ－Ｓｃｈｒｏｄｉｎｇｅｒ方程的Ｃａｕｃｈｙ问题的局部适定性．     

Boundary and initial-value methods for solving Fredholm equations with semidegenerate kernels

We develop a new approach to the theory and numerical solution of a class of linear and nonlinear Fredholm equations. These equations, which have semidegenerate kernels, are shown to be equivalent to two-point boundary-value problems for a system of ordinary differential equations. Applications of numerical methods for this class of problems allows us to develop a new class of numerical algorithms for the original integral equation. The scope of the paper is primarily theoretical; developing the necessary Fredholm theory and giving comparisons with related methods. For convolution equations, the theory is related to that of boundary-value problems in an appropriate Hilbert space. We believe that the results here have independent interest. In the last section, our methods are extended to certain classes of integrodifferential equations.     

Cauchy problem for integrable discrete equations on quad-graphs

Initial value problems for the integrable discrete equations on quad-graphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete KdV equation an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice are discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.Dedicated to S. P. Novikov on his 65 birthdayOn leave from Landau Institute for Theoretical Physics, Chernogolovka, Russia.     

On the local cauchy problem for quasilinear hyperbolic functional differential systems

Generalized solutions of weakly coupled functional differential systems are investigated. Theorems on the existence, uniqueness and continuous dependence upon initial data are given. The local Cauchy problem is transformed into functional integral equations. The method of bicharacteristics and integral inequalities are used.     

Third order homogeneous weakly hyperbolic equations with nonanalytic coefficients

We consider the Cauchy problem for homogeneous linear third order weakly hyperbolic equations with time depending coefficients. We study the relation between the regularity of the coefficients and the Gevrey class in which the Cauchy problem is well-posed.     

Existence and Uniqueness of the Cauchy Problem for a Generalized Navier-Stokes Equations

We consider the Cauchy problem for a generalized Navier-Stokes equations with hyperdissipation, with the initial data in L^p_σ. We follow the theme of [1] but with more complicated analysis on the symbol and obtain the existence and uniqueness results.     

On the Cauchy Problem for a Class of Semilinear Hyperbolic Equations with Discontinuous Coefficients and Data

ln this paper we establish the existence of local solution to the Cauchy problem for a class of semilinear second order hyperbolic equations including degenerated type equations with discontinuous coefficients and data.     

Well-posedness of the cauchy problem for the coupled system of the Schrödinger-KdV equations

This paper is devoted to the study of the Cauchy problem for the coupled system of the Schrödinger-KdV equations which describes the nonlinear dynamics of the one-dimensional Langmuir and ion-acoustic waves. Global well-posedness of the problem is established in the spaceH ^κ ×H ^κ (κ εZ ⁺), the first and second components of which correspond to the electric field of the Langmuir oscillations and the low-frequency density perturbation respectively.     

两类方程Cauchy问题经典解的等价性

对于给出的一类二阶线性双曲型方程,通过未知变量替换,将其化为一阶对称双曲型方程组.可以证明这个一阶对称双曲型方程组与原来的二阶线性双曲型方程的Cauchy问题的经典解在某种意义下是等价的.     

一类非线性Schr(o)dinger方程的整体小解

本文研究带有非线性项|u|~pu的高阶非线性Schr(?)dinger方程的Cauchy问题．对于p的某一取值范围。我们证明了此问题整体解的存在唯一性,并得到了解关于初值的连续依赖性及解具有较强的衰减估计。