Weak solutions for a high-order pseudo-parabolic equation with variable exponents

In this paper, the authors investigate the existence and uniqueness of weak solutions of the initial and boundary value problem for a fourth-order pseudo-parabolic equation with variable exponents of non-linearity. Finally, the authors also obtain a long-time behaviour of weak solutions.     

Well-posedness and dynamics for the fractional Ginzburg-Landau equation

This article studies the global well-posedness and long-time dynamics for the nonlinear complex Ginzburg–Landau equation involving fractional Laplacian. The global existence and some uniqueness criterion of weak solutions are given with compactness method. To study the strong solutions with the semigroup method, we generalize some pointwise estimates for the fractional Laplacian to the complex background and study carefully the linear evolution of the equation. Finally, the existence of global attractors is studied.     

Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media

In this paper, we prove the uniqueness of weak solutions for a pseudo-parabolic equation modeling two-phase flow in a porous medium, where dynamic effects are included in the capillary pressure. We transform the equation into an equivalent system, and then prove the uniqueness of weak solutions to the system which leads to the uniqueness of weak solutions for the original model.     

On variational methods to non-instantaneous impulsive fractional differential equation

In this paper by using the variational methods for a class of impulsive differential equation of fractional order with non-instantaneous impulses, we setup sufficient conditions for the existence and uniqueness of weak solutions. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of a functional. Main results of the present work are established by using Lax–Milgram Theorem.     

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.     

一类退化抛物方程的边值问题

In this paper, we study the boundary value problem (BVP) for a degenerate parabolic equation. By introducing a proper notion of weak solutions, we prove the uniqueness and existence of weak solutions of the problem. The localization of weak solutions is also discussed, which plays a key role in the proof of the uniqueness.     

关于带有三点边值的非线性分数阶积分微分方程的解的存在性和惟一性

本文考虑一个带有三点边值的非线性分数阶积分微分方程,并分别用Krasnoselskii不动点定理和Banach压缩映像原理建立了其解的存在性和惟一性的充分条件.     

First-order differential-operator equation with variable domains of piecewise smooth operators

We prove the existence, uniqueness, and smoothness of weak solutions of a first-order differential-operator equation with variable domains of nonself-adjoint piecewise smooth operators for which one has the corresponding majorant operators. We analyze the well-posedness and smoothness of weak solutions of three new mixed problems with piecewise smooth (in time) coefficients in the equations of finite and infinite order and in the boundary conditions.     

On the solution of a fractional diffusion integrodifferential equation with Rothe time discretization

This article deals with a fractional diffusion equation of the second-order differential Volterra operator and fractional integral condition. Existence and uniqueness of a weak solution in an appropriate sense as well as some regularity results are obtained by the use of Rothe’s method. Finally, an example is given to demonstrate the effectiveness of the proposed approach.     

Existence and uniqueness results for fractional integrodifferential equations with boundary value conditions

In this paper, we study existence and uniqueness of fractional integrodifferential equations with boundary value conditions. A new generalized singular type Gronwall inequality is given to obtain an important a priori bounds. Existence and uniqueness results of solutions are established by virtue of fractional calculus and fixed point method under some weak conditions. An example is given to illustrate the results.     

分数阶微分方程的一些新结果

第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性（次线性）条件下C310,11正解存在的充分必要条件．最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．     

Existence and uniqueness of solutions for a nonlinear fractional differential equation

In this paper we prove the existence and uniqueness of solutions for a class of the nonlinear fractional differential equation with initial condition and investigate the dependence of the solution on the order of the differential equation and on the initial condition. Then we give an example to demonstrate the main results.     

Local existence and uniqueness in the largest critical space for a surface growth model

We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined, which turns out to be a Besov space. Similarly to 3D-Navier Stokes, the uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small.     

Well-posedness for a model of prion proliferation dynamics

The model considered consists of an ordinary differential equation coupled with an integro-partial differential equation and describes the interaction between non-infectious and infectious prion proteins. We provide sufficient conditions for uniqueness of monomer-preserving weak solutions. In addition, we also prove existence of weak solutions under rather general assumptions on the involved degradation rates.     

Blow-up of solutions for a time-space fractional evolution system

In this paper, firstly, we study the local existence and uniqueness of mild solutions for fractional evolution systems with nonlocal in time nonlinearity. Then, we claim that such a mild solution is weak solution of this system. Finally, we prove a blow-up result under some conditions.     

On the Unique Solvability of a Nonlinear Functional Evolution Equation

Existence and uniqueness of weak solutions to an abstract functional evolution equation are proved. An application of this theory to a nonlinear nonlocal reaction-diffusion equation is presented.     

Well-Posedness of the Stochastic Fractional Boussinesq Equation With Lévy Noise

This article is devoted to the well-posedness of the stochastic fractional Boussinesq equation with Lévy noise. The commutator estimates are applied to overcome the difficulty in the convergence since the nonlocal fractional diffusion has lower regularity. Based on stopping time technique, weak convergence method, and monotonicity arguments, the global existence and uniqueness of the weak solution are obtained in a fixed probability space.     

Existence and uniqueness of solutions for a fractional differential equation with multi-point boundary value problems

In this paper, we study the existence and uniqueness solutions of a fractional differential equation with multi-point boundary value problems. By using the fixed point theorems, some new results are established and two examples are given to demonstrate the application of main results.     

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

The initial value problem of a nonlinear fractional differential equation is discussed in this paper. Using the nonlinear alternative of Leray-Schauder type and the contraction mapping principle,we obtain the existence and uniqueness of solutions to the fractional differential equation,which extend some results of the previous papers.     

Weak solutions of a mixed problem in a half-strip for a generalized Kawahara equation

We consider a mixed problem in a half-strip for a generalized Kawahara equation describing long nonlinear waves in inhomogeneous media with weak dispersion. We study the existence and uniqueness of generalized solutions in Sobolev weight spaces.