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Symmetry
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Advances in Mathematical Models and Partial Differential Equations: 2nd Edition
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Advances in Mathematical Models and Partial Differential Equations: 2nd Edition

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 2022

Special Issue Editors

Prof. Dr. Yunhu Wang

E-Mail Website
Guest Editor
College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China
Interests: integral systems
Dr. Manwai Yuen

E-Mail Website
Guest Editor
Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong
Interests: partial differential equations; symmetry reduction; blowup; euler-poisson equations; euler equations with or without coriolis force; camassa-holm equations; navier-stokes equations; Magnetohydrodynamics (MHD); analytical and exact solutions; mathematical methods in fluids; classical cosmology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the study of partial differential equations (PDE), “blow-up” or “singularity” means the breakdown of a system within a finite time. The singularity formation in nonlinear physical systems has attracted the attention of many physics and mathematics researchers because of its physical significance and mathematical challenge. In this regard, a PDE system's lifespan is the maximum time before the solutions exist and are sufficiently smooth.

In the study of PDE, symmetry assumptions or reductions are expected to facilitate the study of the lifespan of the nonlinear partial differential systems. In other words, symmetry is especially useful to analyze simpler cases of some complex systems.

In this Special Issue, we expect that a theoretical or numerical study of the lifespan of nonlinear PDE, can be developed. To contribute to this Special Issue, we expect that the theoretical analysis can establish a sufficient condition on initial data that guarantees that the lifespan of the systems is finite. For the numerical study of the lifespan problem, the maximal existence time must be estimated with significant improvement.

Prof. Dr. Yunhu Wang
Dr. Manwai Yuen
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. Symmetry is an international peer-reviewed open access monthly 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 2400 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.

Keywords

  • partial differential equation
  • mathematical method

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Published Papers (4 papers)

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Research

18 pages, 312 KiB  
Article
Lipschitz and Second-Order Regularities for Non-Homogeneous Degenerate Nonlinear Parabolic Equations in the Heisenberg Group
by Huiying Wang, Chengwei Yu, Zhiqiang Zhang and Yue Zeng
Symmetry 2025, 17(5), 799; https://doi.org/10.3390/sym17050799 - 21 May 2025
Viewed by 295
Abstract
In the Heisenberg group Hn, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form [...] Read more.
In the Heisenberg group Hn, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form ∂tu−∑i=12nXiAi(Xu)=K(x,t,u,Xu), where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For 2≤p≤4, we establish the local Lipschitz regularity (u∈Cloc0,1), with the horizontal gradient satisfying Xu∈Lloc∞; (ii) For 2≤p<3, we establish the local second-order horizontal Sobolev regularity (u∈HWloc2,2), with the second-order horizontal derivative satisfying XXu∈Lloc2. These results solve an open problem proposed by Capogna et al. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)
29 pages, 2593 KiB  
Article
Symmetry and Time-Delay-Driven Dynamics of Rumor Dissemination
by Cunlin Li, Zhuanting Ma, Lufeng Yang and Tajul Ariffin Masron
Symmetry 2025, 17(5), 788; https://doi.org/10.3390/sym17050788 - 19 May 2025
Viewed by 323
Abstract
The dissemination of rumors can lead to significant economic damage and pose a grave threat to social harmony and the stability of people’s livelihoods. Consequently, curbing the dissemination of rumors is of paramount importance. The model in the text assumes that the population [...] Read more.
The dissemination of rumors can lead to significant economic damage and pose a grave threat to social harmony and the stability of people’s livelihoods. Consequently, curbing the dissemination of rumors is of paramount importance. The model in the text assumes that the population is homogeneous in terms of transmission behavior. This homogeneity is essentially a manifestation of translational symmetry. This paper undertakes a thorough examination of the impact of time delay on the dissemination of rumors within social networking services. We have developed a model for rumor dissemination, establishing the positivity and boundedness of its solutions, and identified the existence of an equilibrium point. The study further involved determining the critical threshold of the proposed model, accompanied by a comprehensive examination of its Hopf bifurcation characteristics. In the expression of the threshold R0, the parameters appear in a symmetric form, reflecting the balance between dissemination and suppression mechanisms. Furthermore, detailed investigations were carried out to assess both the localized and global stability properties of the system’s equilibrium states. In stability analysis, the symmetry in the distribution of characteristic equation roots determines the system’s dynamic behavior. Through numerical simulations, we analyzed the potential impacts and theoretically examined the factors influencing rumor dissemination, thereby validating our theoretical analysis. An optimal control strategy was formulated, and three control variables were incorporated to describe the strategy. The optimization framework incorporates a specifically designed cost function that simultaneously accounts for infection reduction and resource allocation efficiency in control strategy implementation. The optimal control strategy proposed in the study involves a comparison between symmetric and asymmetric interventions. Symmetric control measures may prove inefficient, whereas asymmetric control demonstrates higher efficacy—highlighting a trade-off in symmetry considerations for optimization problems. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)
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19 pages, 3215 KiB  
Article
Characteristic Analysis of Local Wave Solutions for the (21)-Dimensional Asymmetric Nizhnik–Novikov–Veselov Equation
by Jingyi Chu, Yaqing Liu, Huining Wu and Manwai Yuen
Symmetry 2025, 17(4), 514; https://doi.org/10.3390/sym17040514 - 28 Mar 2025
Viewed by 238
Abstract
This study investigates the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation, a significant model in nonlinear science, using the Kadomtsev–Petviashvili (KP) hierarchy reduction method. Despite the extensive research on the ANNV equation, a comprehensive exploration of its solutions using the KP hierarchy reduction method is [...] Read more.
This study investigates the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation, a significant model in nonlinear science, using the Kadomtsev–Petviashvili (KP) hierarchy reduction method. Despite the extensive research on the ANNV equation, a comprehensive exploration of its solutions using the KP hierarchy reduction method is lacking. This gap is addressed by identifying constraint conditions that transform a specific KP hierarchy equation into the ANNV equation, thereby enabling the derivation of its Gram determinant solutions. By selecting appropriate τ functions, we obtain breather solutions and analyze their dynamic behavior during wave oscillations. Additionally, lump solutions are derived through long-wave limit analysis, revealing their unique characteristics. This study further explores hybrid solutions that combine breathers and lumps, providing new insights to the interaction between these localized wave phenomena. Our findings enhance the understanding of the ANNV equation’s dynamics and contribute to the broader field of nonlinear wave theory. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)
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19 pages, 292 KiB  
Article
On the Cauchy Problem for the Vlasov-Maxwell-Fokker-Planck System in Low Regularity Space
by Yingzhe Fan and Lihua Tan
Symmetry 2025, 17(1), 100; https://doi.org/10.3390/sym17010100 - 10 Jan 2025
Viewed by 618
Abstract
In this study, we investigate the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system near a global Maxwellian in low regularity space. We establish the existence of global mild solutions to the system by employing the energy method, provided that the perturbative initial data is [...] Read more.
In this study, we investigate the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system near a global Maxwellian in low regularity space. We establish the existence of global mild solutions to the system by employing the energy method, provided that the perturbative initial data is sufficiently small. Moreover, despite the absence of zeroth-order dissipation for the magnetic field, we are able to derive exponential decay estimates for solutions in higher-order regularity space. This is achieved by leveraging the higher-order dissipation properties of the magnetic field, which are deduced from the Maxwell equation. Full article
(This article belongs to the Special Issue Advances in Mathematical Models and Partial Differential Equations: 2nd Edition)
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