Communications in Nonlinear Science and Numerical Simulation最新文献

{"title":"Well-posedness and numerical analysis of an elapsed time model with strongly coupled neural networks","authors":"Mauricio Sepúlveda ,&nbsp;Nicolas Torres ,&nbsp;Luis Miguel Villada","doi":"10.1016/j.cnsns.2025.109144","DOIUrl":"10.1016/j.cnsns.2025.109144","url":null,"abstract":"<div><div>The elapsed time equation is an age-structured model that describes the dynamics of interconnected spiking neurons through the elapsed time since the last discharge, leading to many interesting questions on the evolution of the system from a mathematical and biological point of view. In this work, we deal with the case when the transmission after a spike is instantaneous and the case with a distributed delay that depends on the previous history of the system, which is a more realistic assumption. Since the instantaneous transmission case is known to be ill-posed due to non-uniqueness or jump discontinuities, we establish a criterion for well-posedness to determine when the solution remains continuous in time, through an invertibility condition that improves the existence theory under more relaxed hypothesis on the nonlinearity, including the strongly excitatory case. Inspired in the existence theory, we adapt the classical explicit upwind scheme through a robust fixed-point approach and we prove that the approximation given by this scheme converges to the solution of the nonlinear problem through BV-estimates and we extend the idea to the case with distributed delay. We also show some numerical simulations to compare the behavior of the system in the case of instantaneous transmission with the case of distributed delay under different parameters, leading to solutions with different asymptotic profiles.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109144"},"PeriodicalIF":3.8,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hypergeometric behavior of metal oxide varistors in DC circuit breakers","authors":"Yang Liu ,&nbsp;Zhi Jin(Justin) Zhang ,&nbsp;Kayla Chuong ,&nbsp;Zhiyang Jin ,&nbsp;Alfonso J. Cruz ,&nbsp;Lauren M. Garten ,&nbsp;Lukas Graber","doi":"10.1016/j.cnsns.2025.109158","DOIUrl":"10.1016/j.cnsns.2025.109158","url":null,"abstract":"<div><div>This paper analyzes the formula of current waveforms of metal oxide varistors (MOVs) in dc circuit breaker (DCCB) applications. Firstly, the nonlinear integral equation of a DCCB circuit is solved. DCCB operational metrics derivatives from the solution are taken with respect to MOV nonlinearity coefficients to attain their polarities. Both the solution and the polarities are experimentally validated through DCCB MOV accelerated degradation tests. The solution involves Gauss hypergeometric function, demonstrating a hypergeometric behavior of DCCB MOVs. Polarities of the derivatives also suggest increased nonlinearity coefficients for reliable DCCB design due to reduced MOV energy, charge, and conduction time. The discovery of DCCB MOV hypergeometric behavior and polarities of its metrics illustrates the strength of the hypergeometric model. Besides, multi-disciplinary applications of the model are found, and an asymptotic approximation of the model by the conventional linear model is established.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109158"},"PeriodicalIF":3.8,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Instability and stability analysis of three-dimensional nonhomogeneous incompressible fluid in a spherical ring under the influence of a general potential","authors":"Huichao Wang ,&nbsp;Ruili Wu ,&nbsp;Quan Wang","doi":"10.1016/j.cnsns.2025.109190","DOIUrl":"10.1016/j.cnsns.2025.109190","url":null,"abstract":"&lt;div&gt;&lt;div&gt;This paper investigates the linear and nonlinear stability/instability of three-dimensional nonhomogeneous incompressible viscous fluids in a bounded spherical ring domain &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mstyle&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, subject to Navier-slip boundary conditions and influenced by a general potential &lt;span&gt;&lt;math&gt;&lt;mi&gt;ψ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. This potential is commonly used to model fluid motions in celestial bodies. First, we demonstrate that the system only admits steady-state solutions of the form &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; satisfy the hydrostatic balance condition &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;ψ&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Moreover, the relationship between &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and the potential function &lt;span&gt;&lt;math&gt;&lt;mi&gt;ψ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is constrained by the condition &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;ψ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, enabling us to express &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; as &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;ψ&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Second, when there exists a point &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, we establish the linear instability of these solutions. Furthermore, we demonstrate their nonlinear instability in Hadamard senses through detailed nonlinear energy estimates. This instability aligns with the well-known Rayleigh-Taylor instability. Our study significantly extends and generalizes existing mathematical results, which have predominantly focused on scenarios involving a uniform gravitational field characterized by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;ψ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Finally, we show that these steady-state solutions are linearly stable provided that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; holds throughout the domain. Moreover,","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109190"},"PeriodicalIF":3.8,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Relaxed explicit-Scalar Auxiliary Variable (R-Explicit-SAV) method for hydrodynamically-coupled phase-field crystal model","authors":"Jincheng Pu ,&nbsp;Xiaofeng Yang ,&nbsp;Jun Zhang","doi":"10.1016/j.cnsns.2025.109176","DOIUrl":"10.1016/j.cnsns.2025.109176","url":null,"abstract":"<div><div>In this work, we develop two efficient numerical schemes for solving the hydrodynamically coupled phase-field crystal (PFC) model by combining the explicit-type Scalar Auxiliary Variable (Explicit-SAV) approach with a relaxation-based energy-pulling strategy. Two linear relaxation terms are introduced to penalize deviations in the auxiliary variables, thereby correcting the energy discrepancy introduced by the Explicit-SAV reformulation and enhancing both the accuracy and thermodynamic consistency of the scheme. The resulting time-marching algorithms are linear, fully decoupled, and unconditionally energy-stable. Moreover, by selecting optimal relaxation parameters, we rigorously establish the energy stability of both schemes, demonstrate their unique solvability, and provide detailed descriptions of the decoupling algorithm. Extensive numerical experiments, including shear-induced pattern formation and particle sedimentation, are conducted to validate the accuracy, efficiency, and robustness of the proposed methods.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109176"},"PeriodicalIF":3.8,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the explicit solution for some nonlinear diffusion equations in heterogeneous porous media","authors":"Sebastián Ossandón","doi":"10.1016/j.cnsns.2025.109156","DOIUrl":"10.1016/j.cnsns.2025.109156","url":null,"abstract":"<div><div>In pursuit of understanding heat dynamics within complex structures, in this work, we introduce an approach to find an adequate framework, with an explicit solution, applicable to some non-linear diffusion equations in heterogeneous porous media. The introduced methodology expresses complex solutions as a combination of simpler eigenmodes, allowing a mathematical analysis based on more classical results of semigroup theory and establishing the desired framework for solutions that are applicable to a broad range of physical problems.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109156"},"PeriodicalIF":3.8,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New viscosity-projection methods for solving variational inquality problems with applications to image restoration problems","authors":"C.T.T. Trang ,&nbsp;H.P. Tu ,&nbsp;P.N. Anh","doi":"10.1016/j.cnsns.2025.109179","DOIUrl":"10.1016/j.cnsns.2025.109179","url":null,"abstract":"<div><div>This research investigates two new viscosity-projection methods for solving variational inquality problems <span><math><mrow><mo>(</mo><mi>V</mi><mi>I</mi><mi>P</mi><mi>s</mi><mo>)</mo></mrow></math></span> with applications to image restoration problems via approximation projection in a real Hilbert space. We design the viscosity Mann-type iteration progress accelerated approximation projection rule to solve the pseudomonotone <span><math><mrow><mi>V</mi><mi>I</mi><mi>P</mi><mi>s</mi></mrow></math></span>. Then, we present two strongly convergent algorithms that can be easily implemented, as examples for solving image restoration problems. Numerical experiments illustrate and compare the performances of the proposed algorithms with three other well-known algorithms.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109179"},"PeriodicalIF":3.8,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Oscillatory collision approach in the Earth–Moon restricted three body problem","authors":"Maciej J. Capiński,&nbsp;Aleksander Pasiut","doi":"10.1016/j.cnsns.2025.109173","DOIUrl":"10.1016/j.cnsns.2025.109173","url":null,"abstract":"<div><div>We consider the Earth–Moon planar circular restricted three body problem and present a proof of the existence of orbits, which approach arbitrarily close to one of the primary masses, and at the same time after each approach they move away from the mass to a prescribed distance. In other words the orbits oscillate between being arbitrarily close to collision and away from it. We achieve our goal with the use of topological tools combined with rigorous interval computations. We use the Levi-Civita regularization and validate that the dynamics in the regularized coordinates leads to a good topological alignment between various sets. We then perform shadowing arguments that this leads to the required dynamics in the original coordinates of the system.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109173"},"PeriodicalIF":3.8,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solvability, stability and its application in the P–M synchronization problem of discrete-time fractional order singular systems","authors":"Duong Thi Hong ,&nbsp;Do Duc Thuan","doi":"10.1016/j.cnsns.2025.109175","DOIUrl":"10.1016/j.cnsns.2025.109175","url":null,"abstract":"<div><div>Fractional order singular systems are an important class of systems characterized by algebraic constraints combined with fractional-order dynamic behaviors. This paper focuses on discrete-time fractional order singular systems (DFOSSs), introducing the concept of the index to analyze their structural properties. Using the Drazin inverse, we establish a lemma that decomposes DFOSSs into simpler subsystems, forming the basis for deriving solvability and stability conditions. These results are achieved through techniques from fractional calculus and singular systems. Additionally, we provide an explicit solution formula for DFOSSs, enabling practical computation. A control strategy is then proposed to achieve <span><math><mrow><mi>P</mi><mtext>–</mtext><mi>M</mi></mrow></math></span> synchronization, a method that synchronizes different dimensions within the same master–slave system, surpassing traditional synchronization approaches. To demonstrate the utility of our findings, practical applications in electrical circuits are presented, showcasing the effectiveness of our methods. This study offers a comprehensive framework for analyzing and controlling DFOSSs, bridging theoretical insights with real-world applications.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109175"},"PeriodicalIF":3.8,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144739271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Development of an accurate and robust HLLC-type Riemann solver for all Mach number flows","authors":"Lijun Hu,&nbsp;Kexin Zhu,&nbsp;Lielong Li","doi":"10.1016/j.cnsns.2025.109178","DOIUrl":"10.1016/j.cnsns.2025.109178","url":null,"abstract":"<div><div>Due to its positivity-preserving, entropy condition satisfying and easy extension to other types of hyperbolic equations, the HLLC scheme has become a popular Riemann solver to calculate numerical fluxes. However, there are two drawbacks needed to be tackled before it becomes an impeccable flux solver for various compressible flows: one is the numerical instability in calculating multidimensional strong shock waves; The other is the failure to converge to the desired limit solution in calculating low Mach number flows approaching the incompressible limit. In the current work, the shock instability of the HLLC scheme is cured by simply modifying the nonlinear wave speeds and a hybrid strategy is adopted for accurate calculations of contact waves and rarefaction waves. In addition, the performance of the HLLC scheme in simulating low Mach number flows is improved by controlling the excessive numerical dissipation in momentum equations under low-speed flow regime. The excellent performance of the proposed scheme for simulating flow problems across high and low Mach numbers are demonstrated by a suit of canonical numerical test cases.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109178"},"PeriodicalIF":3.8,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence and regularity of the fractional Keller–Segel equation via the extended theory of semigroups","authors":"Yuting Chen,&nbsp;Zhenbin Fan,&nbsp;Gang Li","doi":"10.1016/j.cnsns.2025.109154","DOIUrl":"10.1016/j.cnsns.2025.109154","url":null,"abstract":"<div><div>This paper intends to look into the existence and regularity of the solution for the fractional Keller–Segel equation by virtue of the extended theory of semigroups, that is, the theory of resolvent families, which is one of the powerful tools for solving partial differential equations. First of all, we establish the theory of resolvent families with two parameters related to the fractional Keller–Segel equation, including the generation theorem and the estimations of resolvent families. Then, we research the existence and Hölder regularity results of the classical solution in fundamental spaces <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> by applying the theory of resolvent families with two parameters, respectively. Moreover, we attain some new <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>-<span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> estimations of the resolvent families by utilizing the Gagliardo–Nirenberg inequality, and investigate the local existence and integrability of the mild solution in fundamental spaces <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. In the end, we explore the global existence of the mild solution in fundamental spaces <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mfrac><mrow><mi>N</mi></mrow><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Compared to the results that derived from the perspective of partial differentiation, we possess some new meaningful results and findings. The results of this article can be served as supplements to those obtained using the methods of partial differentiation.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"152 ","pages":"Article 109154"},"PeriodicalIF":3.8,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}