Special Issue "Modern Analysis and Partial Differential Equation"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (30 June 2020).

Special Issue Editors

Prof. Dr. Peng Cao
E-Mail Website
Guest Editor
Beijing Institute of Technology
Interests: Analysis (functional analysis, operator theory)
Prof. Dr. Junyong Zhang
E-Mail Website
Guest Editor
Beijing Institute of Technology
Interests: Harmonic analysis, Microlocal analysis, Geometric analysis, Partial differential equations
Prof. Dr. Jonathan Ben-Artzi
E-Mail Website
Guest Editor
Cardiff University
Interests: analysis and partial differential equations

Special Issue Information

Dear Colleagues,

Modern analysis, including but not limited to harmonic analysis, functional analysis, microlocal analysis, and geometry analysis is a central topic within mathematical analysis. Growing out of classical Fourier and harmonic analysis, it has developed broadly into many fields of mathematical analysis and partial differential equations, both pure and applied.

This Special Issue, titled “Modern Analysis and Partial Differential Equations,” is designed  to promote the modern analysis method in general, but with a preference for application- oriented papers or survey papers describing concrete aspects of modern analysis and their applications to partial differential equations.

As example topics (though it is not an exhaustive list) let us mention the following areas on Euclidean space or manifolds: operator theory and operator algebra, Fourier Analysis (of functions or distributions), microlocal analysis, harmonic or wavelet analysis, pseudo-differential and Fourier integral operators, dynamic behavior of linear or nonlinear dispersive equations, transport equation and kinetic theory, fluid mechanics, quantum mechanics and more.

Each paper should clearly indicate its focus and motivation for the setting and what kind of impact one may expect from the results presented in this paper.

Prof. Dr. Peng Cao
Prof. Dr. Junyong Zhang
Prof. Dr. Jonathan Ben-Artzi
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (12 papers)

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Research

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Open AccessArticle
The Solvability of a Class of Convolution Equations Associated with 2D FRFT
Mathematics 2020, 8(11), 1928; https://doi.org/10.3390/math8111928 - 02 Nov 2020
Viewed by 446
Abstract
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)
Open AccessArticle
On the Number of Periodic Orbits to Odd Order Differential Delay Systems
by and
Mathematics 2020, 8(10), 1731; https://doi.org/10.3390/math8101731 - 09 Oct 2020
Viewed by 350
Abstract
In this paper, we study the periodic orbits of a type of odd order differential delay system with 2k1 lags via the S1 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 2k1 lags via the S1 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)
Open AccessArticle
p-Laplacian Equations in R + N with Critical Boundary Nonlinearity
Mathematics 2020, 8(9), 1520; https://doi.org/10.3390/math8091520 - 07 Sep 2020
Viewed by 392
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)
Open AccessArticle
Perturbation Theory for Quasinilpotents in Banach Algebras
Mathematics 2020, 8(7), 1163; https://doi.org/10.3390/math8071163 - 15 Jul 2020
Viewed by 414
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)
Open AccessArticle
On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection
Mathematics 2020, 8(7), 1049; https://doi.org/10.3390/math8071049 - 30 Jun 2020
Viewed by 436
Abstract
This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state ( u ¯ = 0 , B ¯ = 0 , 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 ( u ¯ = 0 , B ¯ = 0 , θ ¯ = θ 0 ( y ) ) with the temperature-dependent fluid viscosity, thermal diffusivity and electrical conductivity in a channel. We prove that if a + a , and d 2 d y 2 κ ( θ 0 ( y ) ) 0 or 0 < d 2 d y 2 κ ( θ 0 ( y ) ) 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)
Open AccessArticle
Coupled Systems of Nonlinear Integer and Fractional Differential Equations with Multi-Point and Multi-Strip Boundary Conditions
Mathematics 2020, 8(6), 935; https://doi.org/10.3390/math8060935 - 08 Jun 2020
Viewed by 440
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)
Open AccessArticle
Reflection-Like Maps in High-Dimensional Euclidean Space
Mathematics 2020, 8(6), 872; https://doi.org/10.3390/math8060872 - 28 May 2020
Viewed by 540
Abstract
In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to θ : ( x 1 , x 2 , , x n ) 1 x 1 , x 2 x 1 , , [...] 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 , , x n ) 1 x 1 , x 2 x 1 , , 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 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 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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Open AccessArticle
Logarithmic Decay of Wave Equation with Kelvin-Voigt Damping
Mathematics 2020, 8(5), 715; https://doi.org/10.3390/math8050715 - 03 May 2020
Cited by 1 | Viewed by 450
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)
Open AccessArticle
Boundary Value Problems for a Class of First-Order Fuzzy Delay Differential Equations
Mathematics 2020, 8(5), 683; https://doi.org/10.3390/math8050683 - 01 May 2020
Cited by 1 | Viewed by 439
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)
Open AccessArticle
The Dirichlet Problem of Hessian Equation in Exterior Domains
Mathematics 2020, 8(5), 666; https://doi.org/10.3390/math8050666 - 28 Apr 2020
Viewed by 438
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 det D 2 u = f [...] 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)
Open AccessArticle
Asymptotic Behavior of a Tumor Angiogenesis Model with Haptotaxis
by and
Mathematics 2020, 8(5), 664; https://doi.org/10.3390/math8050664 - 27 Apr 2020
Viewed by 491
Abstract
This paper considers the existence and asymptotic behavior of solutions to the angiogenesis system p t = Δ p ρ · ( p w ) + λ p ( 1 p ) , w t = γ p [...] Read more.
This paper considers the existence and asymptotic behavior of solutions to the angiogenesis system p t = Δ p ρ · ( p w ) + λ p ( 1 p ) , w t = γ p w β in a bounded smooth domain Ω R N ( N = 1 , 2 ) , where ρ , λ , γ > 0 and β 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 , respectively, in L -norm. Full article
(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

Review

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Open AccessReview
A Review on the Qualitative Behavior of Solutions in Some Chemotaxis–Haptotaxis Models of Cancer Invasion
Mathematics 2020, 8(9), 1464; https://doi.org/10.3390/math8091464 - 01 Sep 2020
Viewed by 385
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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