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Special Issue "Modern Analysis and Partial Differential Equations, 2nd Edition"

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

Deadline for manuscript submissions: 29 February 2024 | Viewed by 4874

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Special Issue Editors

Dr. Peng Cao

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Guest Editor

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing Key Lab MCAACI, Beijing 100081, China
Interests: analysis (functional analysis; operator theory)
Special Issues, Collections and Topics in MDPI journals

Dr. Luc Nguyen
E-Mail Website
Guest Editor

Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
Interests: partial differential equations; geometry

Dr. Bo Wang

E-Mail Website
Guest Editor

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
Interests: partial differential equations; nonlinear analysis

Special Issue Information

Dear Colleagues,

Modern analysis, including, but not limited to, harmonic analysis, functional analysis, microlocal analysis, and geometric analysis, is a central topic within mathematical analysis. Growing out of modern analysis, it has developed significantly into many areas, including dynamic systems and differential equations, both pure and applied.

This Special Issue, titled “Modern Analysis and Partial Differential Equations II,” 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.

Topics to be covered include (though this is not an exhaustive list): Operator theory and operator algebra, Harmonic analysis, Nonlinear partial differential equations; Geometric analysis, Dynamic systems, etc.

Each manuscript should clearly indicate its motivation and highlight.

Dr. Peng Cao
Dr. Luc Nguyen
Dr. Bo Wang
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 submissions that pass pre-check are 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 2600 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 (7 papers)

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Research

Open AccessArticle

Determining the Coefficients of the Thermoelastic System from Boundary Information

Xiaoming Tan

Mathematics 2023, 11(9), 2147; https://doi.org/10.3390/math11092147 - 04 May 2023

Viewed by 621

Abstract

Given a compact Riemannian manifold

(M, g)

with smooth boundary

\partial M

, we give an explicit expression for the full symbol of the thermoelastic Dirichlet-to-Neumann map

Λ_{g}

with variable coefficients [...] Read more.

Given a compact Riemannian manifold

(M, g)

with smooth boundary

\partial M

, we give an explicit expression for the full symbol of the thermoelastic Dirichlet-to-Neumann map

Λ_{g}

with variable coefficients

λ, μ, α, β \in C^{\infty} (\bar{M})

. We prove that

Λ_{g}

uniquely determines partial derivatives of all orders of these coefficients on the boundary

\partial M

. Moreover, for a nonempty smooth open subset

Γ \subset \partial M

, suppose that the manifold and these coefficients are real analytic up to

Γ

. We show that

Λ_{g}

uniquely determines these coefficients on the whole manifold

\bar{M}

. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

Open AccessArticle

Large-Time Behavior of Momentum Density Support of a Family of Weakly Dissipative Peakon Equations with Higher-Order Nonlinearity

Xianxian Su

and

Xiaofang Dong

Mathematics 2023, 11(6), 1325; https://doi.org/10.3390/math11061325 - 09 Mar 2023

Cited by 1 | Viewed by 450

Abstract

In this paper, we mainly study the weakly dissipative peakon equations with higher-order nonlinearity. Under the effect of dissipation, we first derive the infinite propagation speed if the initial datum has a nonnegative compact support. Furthermore, we obtain the large-time behavior of the support of momentum density with the initial data compactly supported. The corresponding results are obtained by using some prior estimates and the energy method. It is worth noting that we need to overcome the difficulties caused by the high-order nonlinear structure and the dissipative effect of the equation. The obtained results generalize the previous results to a certain degree. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

Open AccessArticle

Quasilinear Parabolic Equations Associated with Semilinear Parabolic Equations

Katsuyuki Ishii

Michel Pierre

and

Takashi Suzuki

Mathematics 2023, 11(3), 758; https://doi.org/10.3390/math11030758 - 02 Feb 2023

Viewed by 741

Abstract

We formulate a quasilinear parabolic equation describing the behavior of the global-in-time solution to a semilinear parabolic equation. We study this equation in accordance with the blow-up and quenching patterns of the solution to the original semilinear parabolic equation. This quasilinear equation is new in the theory of partial differential equations and presents several difficulties for mathematical analysis. Two approaches are examined: functional analysis and a viscosity solution. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

Open AccessArticle

The Buckling Operator: Inverse Boundary Value Problem

Yanjun Ma

Mathematics 2023, 11(2), 268; https://doi.org/10.3390/math11020268 - 04 Jan 2023

Viewed by 485

Abstract

In this paper, we consider a zeroth-order perturbation

q (x)

of the buckling operator

Δ^{2} - κ Δ

, which can be uniquely determined by measuring the Dirichlet-to-Neumann data on the boundary. We extend the conclusion of the biharmonic operator [...] Read more.

In this paper, we consider a zeroth-order perturbation

q (x)

of the buckling operator

Δ^{2} - κ Δ

, which can be uniquely determined by measuring the Dirichlet-to-Neumann data on the boundary. We extend the conclusion of the biharmonic operator to the buckling operator, but the Dirichlet-to-Neumann map given in this study is more meaningful and general. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

Open AccessArticle

Liouville-Type Theorem for Nonlinear Elliptic Equations Involving Generalized Greiner Operator

Wei Shi

Mathematics 2023, 11(1), 61; https://doi.org/10.3390/math11010061 - 24 Dec 2022

Cited by 1 | Viewed by 530

Abstract

In this paper, we study the Liouville-type behaviors of the generalized Greiner operators with nonlinear boundary value conditions. We use the technique based upon the method of moving planes. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

Open AccessArticle

Positive Radially Symmetric Entire Solutions of p-k-Hessian Equations and Systems

Wei Fan

Limei Dai

and

Bo Wang

Mathematics 2022, 10(18), 3258; https://doi.org/10.3390/math10183258 - 07 Sep 2022

Cited by 2 | Viewed by 835

Abstract

In this paper, we discuss the existence of positive radially symmetric entire solutions of the p-k-Hessian equation [...] Read more.

In this paper, we discuss the existence of positive radially symmetric entire solutions of the p-k-Hessian equation

σ_{k}^{\frac{1}{k}} (λ (D_{i} ({| D u |}^{p - 2} D_{j} u))) = α^{\frac{1}{k}} (| x |) f (u),

and the general p-k-Hessian system

σ_{k}^{\frac{1}{k}} (λ (D_{i} ({| D u |}^{p - 2} D_{j} u))) = α^{\frac{1}{k}} (| x |) f_{1} (v) f_{2} (u)

σ_{k}^{\frac{1}{k}} (λ (D_{i} ({| D v |}^{p - 2} D_{j} v))) = β^{\frac{1}{k}} (| x |) g_{1} (u) g_{2} (v)

. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

Open AccessArticle

Parabolic Hessian Equations Outside a Cylinder

Limei Dai

and

Xuewen Guo

Mathematics 2022, 10(16), 2839; https://doi.org/10.3390/math10162839 - 09 Aug 2022

Viewed by 619

Abstract

In this article, we mainly review the parabolic Hessian equation on the exterior region. The existence and uniqueness of solutions with asymptotic properties to the exterior problem of the parabolic Hessian equation were obtained by using the Perron method. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equations, 2nd Edition)

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Special Issue "Modern Analysis and Partial Differential Equations, 2nd Edition"

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