Initial boundary value problem for generalized logarithmic improved Boussinesq equation

In this paper, we consider the initial boundary value problem for generalized logarithmic improved Boussinesq equation. By using the Galerkin method, logarithmic Sobolev inequality, logarithmic Gronwall inequality, and compactness theorem, we show the existence of global weak solution to the problem. By potential well theory, we show the norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions. Furthermore, for the generalized logarithmic improved Boussinesq equation with damped term, we obtain the decay estimate of the energy. Copyright © 2016 John Wiley & Sons, Ltd.     

Boundary value problem for a global‐in‐time parabolic equation

The aim of this paper is to draw attention to an interesting semilinear parabolic equation that arose when describing the chaotic dynamics of a polymer molecule in a liquid. This equation is nonlocal in time and contains a term, called the interaction potential, that depends on the time‐integral of the solution over the entire interval of solving the problem. In fact, one needs to know the “future” in order to determine the coefficient in this term, that is, the causality principle is violated. The existence of a weak solution of the initial boundary value problem is proven. The interaction potential satisfies fairly general conditions and can have arbitrary growth at infinity. The uniqueness of this solution is established with restrictions on the length of the considered time interval.     

A non-homogeneous boundary-value problem for the KdV–Burgers equation posed on a finite domain

We consider the initial–boundary value problem for the KdV–Burgers equation posed on a bounded interval $[a,b]$. This problem features non-homogeneous boundary conditions applied at $x=a$ and $x=b$ and is known to have unique global smooth solution.     

Initial boundary value problem for a system of generalized IMBq equations

The existence and uniqueness of the global generalized solution and the global classical solution to the initial boundary value problem for a system of generalized IMBq equations are proved. This paper also arrives at some sufficient conditions of blow up of the solution in finite time. Copyright © 2004 John Wiley & Sons, Ltd.     

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s >?2. Copyright © 2017 John Wiley & Sons, Ltd.     

An operator approach for an oblique derivative boundary‐transmission problem

We consider a boundary‐transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain conditions are assumed on it in the form of oblique derivatives. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well‐posedness of the problem is shown for a range of non‐critical regularity orders of the Bessel potential spaces, which include the finite energy norm space. In addition, an operator normalization method is applied to the critical orders case. Copyright © 2004 John Wiley & Sons, Ltd.     

Nonnegative solutions of the characteristic initial value problem for linear partial functional–differential equations of hyperbolic type

On the rectangle , the problem on the existence and uniqueness of a nonnegative solution of the characteristic initial value problem for the equation

is considered, where is a linear bounded operator and .     

Initial boundary value problem for the evolution system of MHD type describing geophysical flow in three‐dimensional domains

The initial boundary value problem for the evolution system describing geophysical flow in three‐dimensional domains was considered. The existence and uniqueness of global strong solution to the evolution system were proved under assumption on smallness of data. Moreover, solvable compatibility conditions of initial data and boundary values which guarantee the existence and uniqueness of global strong solution were discussed. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary‐value problems for the third‐order loaded equation with noncharacteristic type‐change boundaries

The purpose of this paper is to establish unique solvability for a certain generalized boundary‐value problem for a loaded third‐order integro‐differential equation with variable coefficients. Moreover, the method of integral equations is applied to obtain an equation related to the Riemann‐Liouville operators.     

Applications of variational methods to Sturm–Liouville boundary‐value problem for fourth‐order impulsive differential equations

In this paper, we study Sturm–Liouville boundary‐value problem for fourth‐order impulsive differential equations. Applying variational methods, several new existence results are obtained. Copyright © 2013 John Wiley & Sons, Ltd.     

Well‐posedness for the Cauchy problem of the modified Hunter–Saxton equation in the Besov spaces

This paper is concerned with the Cauchy problem of the modified Hunter‐Saxton equation. The local well‐posedness of the model equation is obtained in Besov spaces (which generalize the Sobolev spaces H^s) by using Littlewood‐Paley decomposition and transport equation theory. Moreover, the local well‐posedness in critical case (with ) is considered.     

A note on initial value problem for the generalized Tricomi equation in a mixed-type domain

In this paper, we study the well-posedness of initial value problem for n-dimensional gener-alized Tricomi equation in the mixed-type domain {(t,x):t∈[1,+∞),x∈Rn} with the initial data given on the line t=1 in Hadamard's sense. By taking partial Fourier transformation, we obtain the explicit expression of the solution in terms of two integral operators and further establish the global estimate of such a solution for a class of initial data and source term. Finally, we establish the global solution in time direction for a semilinear problem used the estimate.     

Homogenization of the Signorini boundary‐value problem in a thick junction and boundary integral equations for the homogenized problem

We consider a mixed boundary‐value problem for the Poisson equation in a thick junction Ω_ε which is the union of a domain Ω₀ and a large number of ε—periodically situated thin cylinders. The non‐uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as ε→0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non‐uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as ε→0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non‐standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in Ω₀ and an appropriate postprocessing. The equations in Ω₀ finally are also treated with boundary integral equations. Copyright © 2010 John Wiley & Sons, Ltd.     

The Cauchy problem for the generalized hyperelastic rod wave equation

In this paper we consider the Cauchy problem for the generalized hyperelasticrod wave equation which includes the Camassa‐Holm equation and the hyperelastic rod wave equation. Firstly, by using the Kato's theory, we prove that the Cauchy problem for the generalized hyperelastic rod wave is locally well‐posed in Sobolev spaces with . Secondly, we give some conservation laws, some useful conclusions and the precise blow‐up scenario and show that the Cauchy problem for the generalized hyperelastic rod wave equation has smooth solutions which blows up in finite time. Thirdly, we give the blow‐up rate of the strong solutions to the Cauchy problem for the generalized hyperelastic rod wave equation. Finally, we give the lower bound of the maximal existence time of the solution and the lower semicontinuity of existence time of solutions to the generalized hyperelastic rod wave equation.     

Solvability of the Neumann problem for a Sobolev pseudoparabolic equation

The dynamic potential constructed in this paper is used to analyze the existence of a classical solution to the Neumann problem for a Sobolev equation.     

具p-Laplace算子的分数阶微分方程边值问题的正解

考虑具有p-Laplace算子的分数阶泛函微分方程边值问题,利用锥上的不动点定理,得到了其正解及多个正解存在的充分条件,所得结果推广了已有的结论,并举例说明了结论的适用性.     

Regularity of the solution for a final value problem for the Rayleigh‐Stokes equation

In this paper, we deal with the backward problem of determining initial condition for Rayleigh‐Stokes where the data are given at a fixed time. The problem has many applications in some non‐Newtonian fluids. We give some regularity properties of the solution to backward problem.     

一类分数阶差分方程边值问题多重正解的存在性

考虑一类高阶分数阶差分方程边值问题.构造相关的格林函数,利用不等式技巧,分析格林函数的特征性质.运用不动点指数理论,获得了该分数阶差分方程边值问题存在多重正解的充分条件,举例说明了所获理论的有效性.