The generalized Cauchy problem with data on two surfaces for a quasilinear analytic system

We consider the generalized Cauchy problem with data on two surfaces for a second-order quasilinear analytic system. The distinction of the generalized Cauchy problem from the traditional statement of the Cauchy problem is that the initial conditions for different unknown functions are given on different surfaces: for each unknown function we pose its own initial condition on its own coordinate axis. Earlier, the generalized Cauchy problem was considered in the works of C. Riquier, N. M. Gyunter, S. L. Sobolev, N. A. Lednev, V. M. Teshukov, and S. P. Bautin. In this article we construct a solution to the generalized Cauchy problem in the case when the system of partial differential equations additionally contains the values of the derivatives of the unknown functions (in particular outer derivatives) given on the coordinate axes. The last circumstance is a principal distinction of the problem in the present article from the generalized Cauchy problems studied earlier.     

Generalized boundary value problems for analytic functions with convolutions and its applications

In this paper, we first establish a locality theory for the Noethericity of generalized boundary value problems on the spaces . By means of this theory, of the classical boundary value theory, and of the theory of Fourier analysis, we discuss the necessary and sufficient conditions of the solvability and obtain the general solutions and the Noether conditions for one class of generalized boundary value problems. All cases as regards the index of the coefficients in the equations are considered in detail. Moreover, we apply our theoretical results to the solvability of singular integral equations with variable coefficients. Thus, this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory.     

On analytic solutions to the generalized Cauchy problem with data on three surfaces for a quasilinear system

The generalized Cauchy problem with data on three surfaces is under consideration for a quasilinear analytic system of the third order. Under some simplifying assumption, we find necessary and sufficient conditions for existence of a solution in the form of triple series in the powers of the independent variables. We obtain convenient sufficient conditions under which the data of the generalized Cauchy problem has a unique locally analytic solution. We give counterexamples demonstrating that in the case we study it is impossible to state necessary and sufficient conditions for analytic solvability of the generalized Cauchy problem. We also show that the analytic solution can fail to exist even if the generalized Cauchy problem with data on three surfaces has a formal solution since the series converge only at a sole point, the origin.     

On Cauchy problems of thermal non-equilibrium flows with small data

We study the equations describing the motion of a thermal non-equilibrium gas with one non-equilibriummode. In three space dimensions it is a hyperbolic system of six equations with a relaxation term. The dissipation mechanism induced by the relaxation is weak in the sense that Shizuta-Kawashima criterion is violated. However, there is a significant difference between one dimensional and three dimensional flows in how the criterion is violated. As a consequence, the velocity components in their solutionsbehave differentlywhile thermal dynamic variables share common properties.     

Concerning the Strauss Conjecture and Almost Global Existence for Nonlinear Dirichlet-Wave Equations in 4-Dimensions

We show the obstacle version of the Strauss conjecture holds when the spatial dimension is equal to 4. We also show that an almost global existence theorem of Hörmander for (4 + 1)-dimensional Minkowski space holds in the obstacle setting. We use weighed space-time variants of the energy inequality and a variant of the classical Hardy inequality.     

Global solutions of the evolutionary Faddeev model with small initial data

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space ℝ¹⁺ⁿ to the unit sphere $ \mathbb{S} $ \mathbb{S} ², which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.     

Asymptotics from Scaling for Nonlinear Wave Equations

We present a scaling technique which transforms the evolution problem for a nonlinear wave equation with small initial data to a linear wave equation with a distributional source. The exact solution of the latter uniformly approximates the late-time behavior of solutions of the nonlinear problem in timelike and null directions.     

一类具阻尼非线性波动方程的Cauchy问题

本文研究一类具阻尼非线性波动方程Cauchy问题整体广义解和整体古典解的存在唯一性,并用凸性方法给出解爆破的充分条件.     

复函数Cauchy微分中值定理的推广

以分割区域D为基础将解析函数与共轭解析函数的微分中值定理推广到高阶形式.     

一类非线性耗色散方程解的Blow—up

本文研究一类四阶非线性耗、色散波动方程的补边值问题,在一定条件下,得到了方程解的blow up性质。     

Nonlinear equations with unbounded heat conduction and integrable data

We consider a class of quasi-linear diffusion problems involving a matrix A(t,x,u) which blows up for a finite value m of the unknown u. Stationary and evolution equations are studied for L ¹ data. We focus on the case where the solution u can reach the value m. For such problems we introduce a notion of renormalized solutions and we prove the existence of such solutions.      

Real analytic families of harmonic functions in a planar domain with a small hole

We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ?   and we denote by _u?$u_{?}$ the corresponding solution. The behavior of _u?$u_{?}$ for ?   small and positive can be described in terms of real analytic functions of two variables evaluated at (?,1/log??)$(?,1/\log ??)$. We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by _u?$u_{?}$ for ?   small and describe _u?$u_{?}$ by real analytic functions of ?. Then it is natural to ask what happens when ? is negative. The case of boundary data depending on ? is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.     

Infinite systems of linear equations for real analytic functions

We study the problem when an infinite system of linear functional equations

has a real analytic solution on for every right-hand side and give a complete characterization of such sequences of analytic functionals . We also show that every open set has a complex neighbourhood such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on .

     

A Runge theorem for generalized analytic functions

We show an approximation theorem of Runge type for solutions of the generalized Vekua equation $Lu=Au+B\overline{u}$, where L belongs to a class of degenerate elliptic planar vector fields and $A,B\in L^p$. To prove the theorem, we make use of an integral representation for the solutions of the generalized Vekua equation valid on relatively compact sets. As an application, we study the global solvability of the equation $Lu=Au+B\overline{u}+f$ with $f\in L^p$ and some of its consequences.     

Cauchy Problem for a Generalized Nonlinear Dispersive Equation

The solvability of global smooth solution for the Cauchy problem of a generalized nonlinear dispersive equation is studied by using the continuation method. In addition, the convergences of solution for this problem are also discussed.