Research

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12 pages, 751 KiB

Open AccessArticle

The Solvability of a Class of Convolution Equations Associated with 2D FRFT

by Zhen-Wei Li, Wen-Biao Gao and Bing-Zhao Li

Mathematics 2020, 8(11), 1928; https://doi.org/10.3390/math8111928 - 02 Nov 2020

Cited by 7 | Viewed by 1397

In this paper, the solvability of a class of convolution equations is discussed by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates. Firstly, we generalize the 2D FRFT to the polar coordinates setting. The relationship between 2D FRFT and fractional Hankel [...] Read more.

In this paper, the solvability of a class of convolution equations is discussed by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates. Firstly, we generalize the 2D FRFT to the polar coordinates setting. The relationship between 2D FRFT and fractional Hankel transform (FRHT) is derived. Secondly, the spatial shift and multiplication theorems for 2D FRFT are proposed by using this relationship. Thirdly, in order to analyze the solvability of the convolution equations, a novel convolution operator for 2D FRFT is proposed, and the corresponding convolution theorem is investigated. Finally, based on the proposed theorems, the solvability of the convolution equations is studied. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

19 pages, 303 KiB

Open AccessArticle

On the Number of Periodic Orbits to Odd Order Differential Delay Systems

by Weigao Ge and Lin Li

Mathematics 2020, 8(10), 1731; https://doi.org/10.3390/math8101731 - 09 Oct 2020

Cited by 1 | Viewed by 1086

Abstract

In this paper, we study the periodic orbits of a type of odd order differential delay system with

2 k - 1

lags via the

S^{1}

index theory and the variational method. This type of system has not been studied by others. [...] Read more.

In this paper, we study the periodic orbits of a type of odd order differential delay system with

2 k - 1

lags via the

S^{1}

index theory and the variational method. This type of system has not been studied by others. Our results provide a new and more accurate method for counting the number of periodic orbits. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

24 pages, 329 KiB

Open AccessArticle

p-Laplacian Equations in R + N with Critical Boundary Nonlinearity

by Xu Miao, Junfang Zhao and Xiangqing Liu

Mathematics 2020, 8(9), 1520; https://doi.org/10.3390/math8091520 - 07 Sep 2020

Viewed by 1383

Abstract

In this paper, we consider the following p-Laplacian equation in

R_{+}^{N}

with critical boundary nonlinearity. The existence of infinitely many solutions of the equation is proved via the truncation method. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

7 pages, 238 KiB

Open AccessArticle

Perturbation Theory for Quasinilpotents in Banach Algebras

by Xin Wang and Peng Cao

Mathematics 2020, 8(7), 1163; https://doi.org/10.3390/math8071163 - 15 Jul 2020

Cited by 1 | Viewed by 1446

Abstract

In this paper, we prove the following result by perturbation technique. If q is a quasinilpotent element of a Banach algebra and spectrum of

p + q

for any other quasinilpotent p contains at most n values then

q^{n} = 0

. [...] Read more.

In this paper, we prove the following result by perturbation technique. If q is a quasinilpotent element of a Banach algebra and spectrum of

p + q

for any other quasinilpotent p contains at most n values then

q^{n} = 0

. Applications to C* algebras are given. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

23 pages, 392 KiB

Open AccessArticle

On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection

by Dongfen Bian

Mathematics 2020, 8(7), 1049; https://doi.org/10.3390/math8071049 - 30 Jun 2020

Cited by 2 | Viewed by 1492

Abstract

This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state

(\bar{u} = 0, \bar{B} = 0, \bar{θ} = θ_{0} (y))

with the temperature-dependent [...] Read more.

This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state

(\bar{u} = 0, \bar{B} = 0, \bar{θ} = θ_{0} (y))

with the temperature-dependent fluid viscosity, thermal diffusivity and electrical conductivity in a channel. We prove that if

a_{+} \geq a_{-}

, and

\frac{d^{2}}{d y^{2}} κ (θ_{0} (y)) \leq 0

or

0 < \frac{d^{2}}{d y^{2}} κ (θ_{0} (y)) \leq β_{0}

, with

β_{0} > 0

small enough constant, and then this equilibrium state is nonlinearly asymptotically stable, and if

a_{+} < a_{-}

, this equilibrium state is nonlinearly unstable. Here,

a_{+}

and

a_{-}

are the values of the equilibrium temperature

θ_{0} (y)

on the upper and lower boundary. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

21 pages, 303 KiB

Open AccessArticle

Coupled Systems of Nonlinear Integer and Fractional Differential Equations with Multi-Point and Multi-Strip Boundary Conditions

by Bin Di, Guo Chen and Huihui Pang

Mathematics 2020, 8(6), 935; https://doi.org/10.3390/math8060935 - 08 Jun 2020

Cited by 1 | Viewed by 1568

Abstract

We first consider a second order coupled differential system with nonlinearities involved two unknown functions and their derivatives, subject to a new kinds of multi-point and multi-strip boundary value conditions. Since the coupled system contains two dependent variables and their derivatives, the classical [...] Read more.

We first consider a second order coupled differential system with nonlinearities involved two unknown functions and their derivatives, subject to a new kinds of multi-point and multi-strip boundary value conditions. Since the coupled system contains two dependent variables and their derivatives, the classical method of upper and lower solutions on longer applies. So we adjust and redefine the forms of upper and lower solutions, to establish the existence results. Secondly, we study a Caputo fractional order coupled differential system with discrete multi-point and integral multi-strip boundary value conditions which are very popular recently, and can accurately describe a lot of practical dynamical phenomena, such as control theory, biological system, electroanalytical chemistry and so on. In this part the existence and uniqueness results are achieved via the Leray-Schauder’s alternative and the Banach’s contraction principle. Finally, an example is presented to illustrate the main results. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

11 pages, 676 KiB

Open AccessArticle

Reflection-Like Maps in High-Dimensional Euclidean Space

by Zhiyong Huang and Baokui Li

Mathematics 2020, 8(6), 872; https://doi.org/10.3390/math8060872 - 28 May 2020

Viewed by 1747

Abstract

In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to [...] Read more.

In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to

θ : (x_{1}, x_{2}, \dots, x_{n}) \to (\frac{1}{x_{1}}, \frac{x_{2}}{x_{1}}, \dots, \frac{x_{n}}{x_{1}}) .

We shall prove that reflection-like maps are line-to-line, cross ratios preserving on lines and quadrics preserving. The goal of this article was to consider the rigidity of line-to-line maps on the local domain of

R^{n}

by using reflection-like maps. We mainly prove that a line-to-line map

η

on any convex domain satisfying

η^{\circ 2} = i d

and fixing any points in a super-plane is a reflection or a reflection-like map. By considering the hyperbolic isometry in the Klein Model, we also prove that any line-to-line bijection

f : D^{n} \mapsto D^{n}

is either an orthogonal transformation, or a composition of an orthogonal transformation and a reflection-like map, from which we can find that reflection-like maps are important elements and instruments to consider the rigidity of line-to-line maps. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

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19 pages, 307 KiB

Open AccessArticle

Logarithmic Decay of Wave Equation with Kelvin-Voigt Damping

by Luc Robbiano and Qiong Zhang

Mathematics 2020, 8(5), 715; https://doi.org/10.3390/math8050715 - 03 May 2020

Cited by 6 | Viewed by 1997

Abstract

In this paper, we analyze the longtime behavior of the wave equation with local Kelvin-Voigt Damping. Through introducing proper class symbol and pseudo-diff-calculus, we obtain a Carleman estimate, and then establish an estimate on the corresponding resolvent operator. As a result, we show [...] Read more.

In this paper, we analyze the longtime behavior of the wave equation with local Kelvin-Voigt Damping. Through introducing proper class symbol and pseudo-diff-calculus, we obtain a Carleman estimate, and then establish an estimate on the corresponding resolvent operator. As a result, we show the logarithmic decay rate for energy of the system without any geometric assumption on the subdomain on which the damping is effective. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

9 pages, 265 KiB

Open AccessArticle

Boundary Value Problems for a Class of First-Order Fuzzy Delay Differential Equations

by Hongzhou Wang

Mathematics 2020, 8(5), 683; https://doi.org/10.3390/math8050683 - 01 May 2020

Cited by 2 | Viewed by 1409

Abstract

In this paper, we study a class of fuzzy differential equations with variable boundary value conditions. Applying the upper and lower solutions method and the monotone iterative technique, we provide some sufficient conditions for the existence of solutions, which can be applied to [...] Read more.

In this paper, we study a class of fuzzy differential equations with variable boundary value conditions. Applying the upper and lower solutions method and the monotone iterative technique, we provide some sufficient conditions for the existence of solutions, which can be applied to discuss some dynamical models in biology and economics. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

11 pages, 259 KiB

Open AccessArticle

The Dirichlet Problem of Hessian Equation in Exterior Domains

by Hongfei Li and Limei Dai

Mathematics 2020, 8(5), 666; https://doi.org/10.3390/math8050666 - 28 Apr 2020

Cited by 1 | Viewed by 1308

Abstract

In this paper, we will obtain the existence of viscosity solutions to the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity by the Perron’s method. This extends the Ju–Bao results on Monge–Ampère equations [...] Read more.

In this paper, we will obtain the existence of viscosity solutions to the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity by the Perron’s method. This extends the Ju–Bao results on Monge–Ampère equations

det D^{2} u = f (x)

. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

21 pages, 344 KiB

Open AccessArticle

Asymptotic Behavior of a Tumor Angiogenesis Model with Haptotaxis

by Chi Xu and Yifu Wang

Mathematics 2020, 8(5), 664; https://doi.org/10.3390/math8050664 - 27 Apr 2020

Viewed by 1301

Abstract

This paper considers the existence and asymptotic behavior of solutions to the angiogenesis system [...] Read more.

This paper considers the existence and asymptotic behavior of solutions to the angiogenesis system

p_{t} = Δ p - ρ \nabla \cdot (p \nabla w) + λ p (1 - p), w_{t} = - γ p w^{β}

in a bounded smooth domain

Ω \subset R^{N} (N = 1, 2)

, where

ρ, λ, γ > 0

and

β \geq 1

. More precisely, it is shown that the corresponding solution

(p, w)

converges to

(1, 0)

with an explicit exponential rate if

β = 1

, and polynomial rate if

β > 1

as

t \to \infty

, respectively, in

L^{\infty}

-norm. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

Review

Jump to: Research

11 pages, 254 KiB

Open AccessReview

A Review on the Qualitative Behavior of Solutions in Some Chemotaxis–Haptotaxis Models of Cancer Invasion

by Yifu Wang

Mathematics 2020, 8(9), 1464; https://doi.org/10.3390/math8091464 - 01 Sep 2020

Cited by 4 | Viewed by 1554

Abstract

Chemotaxis is an oriented movement of cells and organisms in response to chemical signals, and plays an important role in the life of many cells and microorganisms, such as the transport of embryonic cells to developing tissues and immune cells to infection sites. [...] Read more.

Chemotaxis is an oriented movement of cells and organisms in response to chemical signals, and plays an important role in the life of many cells and microorganisms, such as the transport of embryonic cells to developing tissues and immune cells to infection sites. Since the pioneering works of Keller and Segel, there has been a great deal of literature on the qualitative analysis of chemotaxis systems. As an important extension of the Keller–Segel system, a variety of chemotaxis–haptotaxis models have been proposed in order to gain a comprehensive understanding of the invasion–metastasis cascade. From a mathematical point of view, the rigorous analysis thereof is a nontrivial issue due to the fact that partial differential equations (PDEs) for the quantities on the macroscale are strongly coupled with ordinary differential equations (ODEs) modeling the subcellular events. It is the goal of this paper to describe recent results of some chemotaxis–haptotaxis models, inter alia macro cancer invasion models proposed by Chaplain et al., and multiscale cancer invasion models by Stinner et al., and also to introduce some open problems. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

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