The Cauchy Problem for a Higher Order Telegraph Equation

Using the integral operator that defines a solution of the Cauchy problem for the equation

Cauchy Problem to a Type of Second Order Differential-iterative Equation

Abstract The paper studies the behavior and existence of strong solutions to the second or-der functional-iterative differential equation     

一类二阶迭代泛函微分方程的初值问题(英文)

本文利用Schauder不动点定理,首次研究了一类二阶迭代泛函微分方程x″(t)= x(x(t)),满足初始条件:x′(σ)= 0;x(σ)= σ的强解的存在性及其性态.     

The Cauchy Problem for the Fifth Order Shallow Water Equation

The local well-posedness of the Cauchy problem for the fifth order shallow water equation δtu＋αδx^5u＋βδx^3u＋rδxu＋μuδru=0,x,t∈R, is established for low regularity data in Sobolev spaces H^s（s≥-3/8） by the Fourier restriction norm method. Moreover, the global well-posedness for L^2 data follows from the local well-posedness and the conserved quantity. For data in H^s（s〉0）, the global well-posedness is also proved, where the main idea is to use the generalized bilinear estimates associated with the Fourier restriction norm method to prove that the existence time of the solution only depends on the L^2 norm of initial data.     

一类广义Boussinesq型方程的Cauchy问题

证明了一类广义Boussinesq型方程Cauchy问题整体解的存在性与唯一性,并给出解在有限时刻爆破的充分条件.     

On the Cauchy Problem for the Third Order Benjamin-Ono Equation

The existence and uniqueness of global solutions to the Cauchyproblem for the third order Benjamin-Ono equation are established.In addition, we obtain an Iorio type result for the asymptoticbehaviour of its solutions.     

Cauchy Problem for Quasilinear Hyperbolic Systems with Higher Order Dissipative Terms

Abstract In this paper, the author studies the global existence, singularities and life span of smoothsolutions of the Cauchy probleth for a class of quasilinear hypetbolic systems with higher order dissipativeterms and gives their applications to nonlinear wave equations with higher order dissipative terms.     

柯西型积分的高阶导数公式证明

本文给出一个较为简单的方法证明柯西型积分的高阶导数公式。     

The Cauchy Problem for a Differential Inclusion in Banach Spaces and Distribution Spaces

We study well-posedness of degenerate Cauchy problems treated as Cauchy problems for a differential inclusion with a multivalued linear operator. Using a new approach to the definition of degenerate integrated semigroups and their generators in a Banach space, we obtain a well-posedness criterion for the problem. Moreover, we consider the Cauchy problem for a differential inclusion in the space of abstract distributions and give necessary and sufficient conditions for well-posedness in the distribution space.     

一类弱耗散Camassa-Holm 方程Cauchy问题在Besov 空间解的局部适定性

利用Littlewood-Paley 理论和输运方程解的先验估计, 在Besov 空间 中证明了一类弱耗散Camassa-Holm 方程Cauchy 问题解的局部适定性, 同时给出了解的能量估计及爆破准则.     

An Operator Approach to the Cauchy Problem for the Euler-Poisson-Darboux Equation in Spaces of Constant Curvature

We develop an operator approach to the problem named in the title. It is based on operator treatment of some relations between Bessel functions, and hypergeometric functions as well. Our operator version of the Delsarte- Povzner formulas also plays an essential role in the paper.     

On Cauchy Problem for First Order Nonlinear Functional Differential Equations of Non-Volterra's Type

On the segment I = [a, b] consider the problem ú(t)=f(u)(t), u(a)=c, where f: C(I, )rightarrow L(I, ) is a continuous, in general nonlinear operator satisfying Carathéodory condition, and c . The effective sufficient conditions guaranteeing the solvability and unique solvability of the considered problem are established. Examples verifying the optimality of obtained results are given, as well.     

The Cauchy Problem for the Generalized Korteweg-de Vries-Burgers Equation in _H

The Cauchy problem for the generalized Korteweg-de Vries-Burgers equation is considered and the local existence and uniqueness of solutions in L^q(0, T;L^p) ∩ L^∞(0, T; \dot{H}^{-s})(0 ≤ s < 1) are obtained for initial data in \dot{H}^{-s}. Moreover, the local solutions are global if the initial data are sufficiently small in critical case. Particularly, for s = 0, the generalized Korteweg-de Vries-Burgers equation satisfies the energy equality, so the initial data can be arbitrarily large to obtain the global solution.     

The Cauchy Problem of a Shallow Water Equation

We consider the Cauchy problem of a shallow water equation and its local wellposedness.     

Kaup-Kupershmidt方程Cauchy问题的局部可解性

This paper deals with the local solvability of initial value problem for Kaup-Kupershmidt equations. Indeed, using Bourgain method, we prove that the Cauchy problem of Kaup-Kupershmidt equation is local well-posed in H8 whenever s 〉 9/8, which improves the former results in [5].     

一类N维非线性波动方程的Cauchy问题

证明下列非线性波动方程的Cauchy问题v_(tt)-α△v_(tt)-Δv=g(v)-αΔg(v),x∈R~N,t>0,(1)v(x,0)=v_0(x),v_t(x,0)=v_1(x),x∈R~N(2)在空间C~2([0,∞);H~s(R~N))(s>N/2)中存在唯一整体广义解v和在空间C~2([0,∞);H~s(R~N))(s>2+N/2N)中存在唯一整体古典解v,即u∈C~2([0,∞);C_B~2(R~N)).还证明Cauchy问题(1),(2)在C~3([0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m≥0,1≤p≤∞)中有唯一整体广义解v和在C~3([0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m>2+N/P)中有唯一整体古典解v,即v∈C~3([0,∞);C~2(R~N)∩L~∞(R~N)).     

Stochastic Regularization Method for Backward Cauchy Problem in Banach Spaces

We consider a stochastic regularization method for solving the backward Cauchy problem in Banach spaces. An order of convergence is obtained on sourcewise representative elements.