Regularity results for fractional diffusion equations involving fractional derivative with Mittag–Leffler kernel

This paper studies partial differential equation model with the new general fractional derivatives involving the kernels of the extended Mittag–Leffler type functions. An initial boundary value problem for the anomalous diffusion of fractional order is analyzed and considered. The fractional derivative with Mittag–Leffler kernel or also called Atangana and Baleanu fractional derivative in time is taken in the Caputo sense. We obtain results on the existence, uniqueness, and regularity of the solution.     

Investigation of electromagnetic wave diffraction from an inhomogeneous partially shielded solid

Diffraction of electromagnetic wave from a partially shielded inhomogeneous dielectric is considered. The original boundary value problem for Maxwell’s equations is shown to have at most one quasi-classical solution. The problem is reduced to a system of integro-differential equations on the solid and the screens. The matrix integro-differential operator is treated in Sobolev spaces and is shown to be a continuously invertible operator. As a result, convergence of the Galerkin method is proved in the chosen functional spaces.     

Analysis of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with p‐Laplacian in Banach space

This paper deals with 2 core aspects of fractional calculus including existence of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with nonlinear p‐Laplacian operator in Caputo sense. For these aims, the suggested problem is converted into an integral equation via Green function , for ε∈(n−1,n], where n≥4. Then, the Green function is examined whether it is increasing or decreasing and positive or negative function. After these properties, some classical fixed‐point theorems are used for the existence of positive solution. Hyers‐Ulam stability of the proposed problem is also considered. For the application of the results, an expressive example is included.     

On the well‐posedness of a Volterra equation with applications in the Navier‐Stokes problem

This paper is dedicated to the study of a family of nonlinear Volterra equations coming from the theory of viscoelasticity. We analyze the existence of local mild solutions to the problem and their possible continuation to a maximal interval of existence. We also consider the problem of continuous dependence with respect to initial data.     

THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton(gμHS2) system with viscous terms.Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative(gμHS2) system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative(gμHS2) system, we show that the semi-group of the solution operator {S(t)}t≥0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S(t)}t≥0 possesses a global attractor in the Sobolev space H~2(S) × H~2(S).     

Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

This paper is concerned with the two‐species chemotaxis‐competition system where Ω is a bounded domain in with smooth boundary ∂Ω, n≥2; χ_i and μ_i are constants satisfying some conditions. The above system was studied in the cases that a₁,a₂∈(0,1) and a₁>1>a₂, and it was proved that global existence and asymptotic stability hold when are small. However, the conditions in the above 2 cases strongly depend on a₁,a₂, and have not been obtained in the case that a₁,a₂≥1. Moreover, convergence rates in the cases that a₁,a₂∈(0,1) and a₁>1>a₂ have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all a₁,a₂>0 which covers the case that a₁,a₂≥1, and lead to convergence rates for solutions of the above system in the cases that a₁,a₂∈(0,1) and a₁≥1>a₂.     

On Erdélyi–Kober Quadratic Functional-Integral Equation in Banach Algebra

Using the technique of a suitable measure of noncompactness in Banach algebra, we prove an existence theorem for some functional-integral equations in Banach algebra, namely Erdélyi–Kober quadratic functional-integral equation. This equation contains as particular cases a lot of integral equations that appear in many branches of nonlinear analysis and its applications. The obtained result is illustrated by an example.     

一类复合型奇异三阶三点边值问题正解的存在性

考察了一类复合型非线性三阶三点边值问题的正解,其中非线性项f(t,u)可以在t=0,t=1及u=0处奇异.利用锥压缩与锥拉伸型的Krasnosel'skii不动点定理建立了几个正解存在定理.当f(t,u)超线性和次线性时,这些存在定理推广了现有的结论.     

The Equation Δu+∇φ•∇u=8πc(1-heu) on a Riemann Surface

Let M be a compact Riemann surface, h(x) a positive smooth function on M, and f(x) a smooth function on M which satisfies that $∫_Me^φdV_g=1$. In this paper, we consider the functional $J(u)=½∫_M|∇u|^2e^φdV_g+8πc∫_Mue^φdV_g-8πclog∫_Mhe^{u+φ}dV_g$. We give a sufficient condition under which J achieves its minimum for $c≤inf_{x∈M^{e^φ(x)}}$.     

抽象空间内中立型随机泛函微分方程解的存在唯一性(英文)

在非利普希茨条件和线性增长条件下,研究了中立型随机泛函微分方程解的存在唯一性,其初始值定义在抽象空间B((-∞,0]; Rd)内.该方程的解是通过皮卡逐步逼近的方法建立的.