Pub Date : 2026-10-01Epub Date: 2026-01-20DOI: 10.1016/j.nonrwa.2026.104601
Lennon Ó Náraigh , Khang Ee Pang , Richard J. Smith
The Geometric Thin-Film Equation is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We show that the weak formulation of the problem, given initial Radon data, admits solutions that are globally defined for all time and are expressible as push-forwards of Borel measurable functions whose behaviour is governed by a set of ordinary differential equations (ODEs). The existence is first demonstrated in the special case of a finite weighted sum of delta functions whose centres evolve over time – these are known as ‘particle solutions’. In the general case, we construct a convergent sequence of particle solutions whose limit yields a solution of the above form. Moreover, we demonstrate that all weak solutions constructed in this way are 1/2-Hölder continuous in time and are uniquely determined by the initial conditions.
{"title":"Convergence analysis of the geometric thin-film equation","authors":"Lennon Ó Náraigh , Khang Ee Pang , Richard J. Smith","doi":"10.1016/j.nonrwa.2026.104601","DOIUrl":"10.1016/j.nonrwa.2026.104601","url":null,"abstract":"<div><div>The Geometric Thin-Film Equation is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We show that the weak formulation of the problem, given initial Radon data, admits solutions that are globally defined for all time and are expressible as push-forwards of Borel measurable functions whose behaviour is governed by a set of ordinary differential equations (ODEs). The existence is first demonstrated in the special case of a finite weighted sum of delta functions whose centres evolve over time – these are known as ‘particle solutions’. In the general case, we construct a convergent sequence of particle solutions whose limit yields a solution of the above form. Moreover, we demonstrate that all weak solutions constructed in this way are 1/2-Hölder continuous in time and are uniquely determined by the initial conditions.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104601"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146037630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-01-31DOI: 10.1016/j.nonrwa.2026.104607
Jaume Llibre , Angela C.T. Sánchez , Durval J. Tonon
A hybrid dynamical system is one whose behavior is governed by both continuous and discrete dynamics; that is, it exhibits both flows and jumps. The field of hybrid dynamical systems is relatively recent and encompasses a broad range of phenomena, and is often used to model various natural processes. In this paper, we investigate the maximum number of limit cycles that can arise in certain classes of discontinuous piecewise differential systems. These systems consist of two hybrid rigid subsystems separated by a straight line, where each rigid subsystem is composed of a linear center perturbed by a homogeneous polynomial of degree 2, 3, 4, 5 or 6. For these classes of piecewise systems, we address the extended 16th Hilbert problem.
{"title":"Limit cycles of discontinuous piecewise hybrid rigid systems separated by a straight line","authors":"Jaume Llibre , Angela C.T. Sánchez , Durval J. Tonon","doi":"10.1016/j.nonrwa.2026.104607","DOIUrl":"10.1016/j.nonrwa.2026.104607","url":null,"abstract":"<div><div>A hybrid dynamical system is one whose behavior is governed by both continuous and discrete dynamics; that is, it exhibits both flows and jumps. The field of hybrid dynamical systems is relatively recent and encompasses a broad range of phenomena, and is often used to model various natural processes. In this paper, we investigate the maximum number of limit cycles that can arise in certain classes of discontinuous piecewise differential systems. These systems consist of two hybrid rigid subsystems separated by a straight line, where each rigid subsystem is composed of a linear center perturbed by a homogeneous polynomial of degree 2, 3, 4, 5 or 6. For these classes of piecewise systems, we address the extended 16th Hilbert problem.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104607"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146188909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-01-27DOI: 10.1016/j.nonrwa.2026.104606
Chun Wu
This paper deals with the following quasilinear chemotaxis systemunder the homogeneous Neumann boundary condition in with smooth boundary ∂Ω, where the parameters a, b > 0 and m > 1. It is shown that there is at least one global weak solution for the system being discussed.
{"title":"Global existence of weak solutions to a quasilinear parabolic chemotaxis system","authors":"Chun Wu","doi":"10.1016/j.nonrwa.2026.104606","DOIUrl":"10.1016/j.nonrwa.2026.104606","url":null,"abstract":"<div><div>This paper deals with the following quasilinear chemotaxis system<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>=</mo><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>∇</mi><msup><mi>u</mi><mi>m</mi></msup><mo>−</mo><mi>u</mi><mi>v</mi><mi>∇</mi><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>a</mi><mi>u</mi><mo>−</mo><mi>b</mi><msup><mi>u</mi><mn>2</mn></msup><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>v</mi><mi>t</mi></msub><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>v</mi><mo>−</mo><mi>u</mi><mi>v</mi><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mtd></mtr></mtable></mrow></math></span></span></span>under the homogeneous Neumann boundary condition in <span><math><mrow><mstyle><mi>Ω</mi></mstyle><mo>⊂</mo><msup><mi>R</mi><mi>n</mi></msup><mspace></mspace><mspace></mspace><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> with smooth boundary ∂Ω, where the parameters <em>a, b</em> > 0 and <em>m</em> > 1. It is shown that there is at least one global weak solution for the system being discussed.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104606"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2025-12-30DOI: 10.1016/j.nonrwa.2025.104582
Shri Harine P, Ankit Kumar
Classical prey-predator models often assume that the predator’s handling time is constant. However, in real ecosystems, a predator’s handling time can vary due to several biotic and abiotic factors. Based on this, we modified the Holling Type II functional response by incorporating a nonlinear handling time function. Fear in prey can lead to notable population reductions, predominantly through decreased foraging and reproduction. Considering these essential factors, we developed a prey-predator model encompassing temporal dynamics, self-diffusion and cross-diffusion. For the temporal model, we investigated the non-negativity, boundedness, and stability conditions of the existing steady states. Furthermore, bifurcations such as Hopf, transcritical, and Bautin were observed with respect to parameters like the cost of fear and the maximal achievable handling time. Bistability behaviour was observed through the analysis involving these two parameters. Sensitivity analysis was conducted to understand the influence of parameters contributing to the coexistence of prey and predator populations. Stability conditions for both spatiotemporal models (with self and cross-diffusion) were established, highlighting the role of cross-diffusion coefficients in inducing Turing instability and pattern formation. Spatial patterns such as spots and vertically aligned chains were observed. An increase in the maximal achievable handling time was found to support prey occupation in high-density regions, promoting coexistence, whereas excessively high maximal handling time can lead to predator extinction.
{"title":"Modelling spatiotemporal prey-predator interactions incorporating fear effect and variable handling time","authors":"Shri Harine P, Ankit Kumar","doi":"10.1016/j.nonrwa.2025.104582","DOIUrl":"10.1016/j.nonrwa.2025.104582","url":null,"abstract":"<div><div>Classical prey-predator models often assume that the predator’s handling time is constant. However, in real ecosystems, a predator’s handling time can vary due to several biotic and abiotic factors. Based on this, we modified the Holling Type II functional response by incorporating a nonlinear handling time function. Fear in prey can lead to notable population reductions, predominantly through decreased foraging and reproduction. Considering these essential factors, we developed a prey-predator model encompassing temporal dynamics, self-diffusion and cross-diffusion. For the temporal model, we investigated the non-negativity, boundedness, and stability conditions of the existing steady states. Furthermore, bifurcations such as Hopf, transcritical, and Bautin were observed with respect to parameters like the cost of fear and the maximal achievable handling time. Bistability behaviour was observed through the analysis involving these two parameters. Sensitivity analysis was conducted to understand the influence of parameters contributing to the coexistence of prey and predator populations. Stability conditions for both spatiotemporal models (with self and cross-diffusion) were established, highlighting the role of cross-diffusion coefficients in inducing Turing instability and pattern formation. Spatial patterns such as spots and vertically aligned chains were observed. An increase in the maximal achievable handling time was found to support prey occupation in high-density regions, promoting coexistence, whereas excessively high maximal handling time can lead to predator extinction.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104582"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2025-12-25DOI: 10.1016/j.nonrwa.2025.104572
Guanghui Wang , Mingying Zhong
The pointwise estimates of the Green’s function for the incompressible Navier–Stokes–Maxwell system with Ohm’s law in 3D are given in this paper. It is shown that the Green’s function consists of the heat kernels, the diffusive waves at low-frequency, the hyperbolic waves at high-frequency with time decaying exponentially, and the singular short waves. In addition, we establish the pointwise estimate of the global solution to the nonlinear incompressible Navier–Stokes–Maxwell system with Ohm’s law based on the Green’s function. To solve the new problem that the nonlinear terms contain the nonlocal operators and which arise from the fluid-electromagnetic decomposition, we develop some new estimates of the nonlocal operators.
{"title":"The pointwise estimates for the incompressible Navier–Stokes–Maxwell system with Ohm’s law","authors":"Guanghui Wang , Mingying Zhong","doi":"10.1016/j.nonrwa.2025.104572","DOIUrl":"10.1016/j.nonrwa.2025.104572","url":null,"abstract":"<div><div>The pointwise estimates of the Green’s function for the incompressible Navier–Stokes–Maxwell system with Ohm’s law in 3D are given in this paper. It is shown that the Green’s function consists of the heat kernels, the diffusive waves at low-frequency, the hyperbolic waves at high-frequency with time decaying exponentially, and the singular short waves. In addition, we establish the pointwise estimate of the global solution to the nonlinear incompressible Navier–Stokes–Maxwell system with Ohm’s law based on the Green’s function. To solve the new problem that the nonlinear terms contain the nonlocal operators <span><math><mrow><mi>∇</mi><mrow><mrow><mi>d</mi></mrow><mi>i</mi><mi>v</mi></mrow><mstyle><mi>Δ</mi></mstyle><msup><mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> and <span><math><mrow><mi>∇</mi><mo>×</mo><mi>∇</mi><mo>×</mo><mstyle><mi>Δ</mi></mstyle><msup><mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> which arise from the fluid-electromagnetic decomposition, we develop some new estimates of the nonlocal operators.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104572"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-01-30DOI: 10.1016/j.nonrwa.2026.104611
Rongrong Yan , Bin Guo , Xiangyu Zhu
In this paper, we consider an initial boundary value problem for the following Timoshenko equation with variable exponents:
First of all, we combine the truncation method, energy estimate method and Banach fixed point theorem as well as Galerkin method to prove the existence of local solutions with the exponent q(x) satisfying Subsequently, for the supercritical case(), owing to the failure of the embedding inequality, the well-known multiplier technique is unsuccessful in our problem. To end this, our strategy is to give a priori estimate for the weighted integral , and then to apply modified weighted multiplier method and potential well method to prove that the energy functional decays logarithmically under this condition. In particular, these results reveal the explicit relationship between decay rate of solutions and the weak damping term. These results improved and extended the existing results [1, 2].
本文考虑了下述变指数Timoshenko方程的初边值问题：utt+Δ2u−M（∥∇u∥2）Δu+| but | M(x)−2ut=|u|q(x)−2u。首先，结合截断法、能量估计法和Banach不动点定理以及Galerkin方法，证明了指数q(x)满足2(n−2)n−4<q(x)<；2nn−4的局部解的存在性。随后，对于超临界情况（m(x)>2nn−4），由于嵌入不等式的失效，众所周知的乘子技术在我们的问题中是不成功的。为此，我们的策略是对加权积分∫Ω（2+t）1−m(x)|u|m(x)dx进行先验估计，然后应用改进的加权乘数法和势阱法证明能量泛函在这种情况下呈对数衰减。特别地，这些结果揭示了解的衰减率与弱阻尼项之间的显式关系。这些结果是对已有结果的改进和扩展[1,2]。
{"title":"Existence and decay of solutions for Timoshenko-type equation with variable exponents and the supercritical damping","authors":"Rongrong Yan , Bin Guo , Xiangyu Zhu","doi":"10.1016/j.nonrwa.2026.104611","DOIUrl":"10.1016/j.nonrwa.2026.104611","url":null,"abstract":"<div><div>In this paper, we consider an initial boundary value problem for the following Timoshenko equation with variable exponents:<span><span><span><math><mrow><msub><mi>u</mi><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><msup><mstyle><mi>Δ</mi></mstyle><mn>2</mn></msup><mspace></mspace><mi>u</mi><mo>−</mo><msup><mrow><mi>M</mi><mo>(</mo><mo>∥</mo><mi>∇</mi><mi>u</mi><mo>∥</mo></mrow><mn>2</mn></msup><mrow><mo>)</mo><mstyle><mi>Δ</mi></mstyle><mi>u</mi><mo>+</mo><mo>|</mo></mrow><msub><mi>u</mi><mi>t</mi></msub><msup><mrow><mo>|</mo></mrow><mrow><mi>m</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mn>2</mn></mrow></msup><msub><mi>u</mi><mi>t</mi></msub><mo>=</mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>.</mo></mrow></math></span></span></span></div><div>First of all, we combine the truncation method, energy estimate method and Banach fixed point theorem as well as Galerkin method to prove the existence of local solutions with the exponent <em>q</em>(<em>x</em>) satisfying <span><math><mrow><mfrac><mrow><mn>2</mn><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo><</mo><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo>.</mo></mrow></math></span> Subsequently, for the supercritical case(<span><math><mrow><mi>m</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>></mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>4</mn></mrow></mfrac></mrow></math></span>), owing to the failure of the embedding inequality, the well-known multiplier technique is unsuccessful in our problem. To end this, our strategy is to give a priori estimate for the weighted integral <span><math><mstyle><mrow><msub><mo>∫</mo><mstyle><mi>Ω</mi></mstyle></msub><msup><mrow><mo>(</mo><mn>2</mn><mo>+</mo><mi>t</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>m</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></msup><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>m</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></msup><mrow><mi>d</mi></mrow><mi>x</mi></mrow></mstyle></math></span>, and then to apply modified weighted multiplier method and potential well method to prove that the energy functional decays logarithmically under this condition. In particular, these results reveal the explicit relationship between decay rate of solutions and the weak damping term. These results improved and extended the existing results [1, 2].</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104611"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146077474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2025-12-24DOI: 10.1016/j.nonrwa.2025.104580
Leander Claes , Michael Winkler
<div><div>In bounded <em>n</em>-dimensional domains with <em>n</em> ≥ 1, this manuscript examines an initial-boundary value problem for the system<span><span><span><math><mtable><mtr><mtd><mrow><mo>{</mo><mtable><mtr><mtd><mrow><msub><mi>u</mi><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>=</mo><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>γ</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mi>∇</mi><msub><mi>u</mi><mi>t</mi></msub><mo>)</mo></mrow><mo>+</mo><mi>a</mi><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>γ</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mi>∇</mi><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>∇</mi><mo>·</mo><mi>f</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mstyle><mi>Θ</mi></mstyle><mi>t</mi></msub><mo>=</mo><mi>D</mi><mstyle><mi>Δ</mi></mstyle><mstyle><mi>Θ</mi></mstyle><mo>+</mo><mstyle><mi>Γ</mi></mstyle><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><msup><mrow><mo>|</mo><mi>∇</mi><msub><mi>u</mi><mi>t</mi></msub><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mo>·</mo><mi>∇</mi><msub><mi>u</mi><mi>t</mi></msub><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></math></span></span></span>which in the case <span><math><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow></math></span> and with <em>γ</em> ≡ Γ as well as <em>f</em> ≡ <em>F</em> reduces to the classical model for the evolution of displacement and temperatures in thermoviscoelasticity. Unlike in previous related studies, the focus here is on situations in which besides <em>f</em> and <em>F</em>, also the core ingredients <em>γ</em> and Γ may depend on the temperature variable Θ. Firstly, a statement on local existence of classical solutions is derived for arbitrary <em>a</em> > 0, <em>D</em> > 0 as well as 0 < <em>γ</em> ∈ <em>C</em><sup>2</sup>([0, ∞)) and 0 ≤ Γ ∈ <em>C</em><sup>1</sup>([0, ∞)), for functions <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>;</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>F</mi><mo>∈</mo><msup><mi>C</mi><mn>1</mn></msup><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>;</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>F</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></mrow></math></span>, and for suitably regular initial data of arbitrary size. Secondly, it is seen that under an additional assumption on smallness of <em>a, f</em>′ and <em>F</em>, as well as on the deviation of the initial data from the constant state given by <span><math><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mstyle><mi>Θ</mi></mstyle><mo>=</mo><msub><mstyle><mi>Θ</mi></mstyle><mi>★</mi>
{"title":"Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W1,p energy analysis","authors":"Leander Claes , Michael Winkler","doi":"10.1016/j.nonrwa.2025.104580","DOIUrl":"10.1016/j.nonrwa.2025.104580","url":null,"abstract":"<div><div>In bounded <em>n</em>-dimensional domains with <em>n</em> ≥ 1, this manuscript examines an initial-boundary value problem for the system<span><span><span><math><mtable><mtr><mtd><mrow><mo>{</mo><mtable><mtr><mtd><mrow><msub><mi>u</mi><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>=</mo><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>γ</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mi>∇</mi><msub><mi>u</mi><mi>t</mi></msub><mo>)</mo></mrow><mo>+</mo><mi>a</mi><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>γ</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mi>∇</mi><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>∇</mi><mo>·</mo><mi>f</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mstyle><mi>Θ</mi></mstyle><mi>t</mi></msub><mo>=</mo><mi>D</mi><mstyle><mi>Δ</mi></mstyle><mstyle><mi>Θ</mi></mstyle><mo>+</mo><mstyle><mi>Γ</mi></mstyle><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><msup><mrow><mo>|</mo><mi>∇</mi><msub><mi>u</mi><mi>t</mi></msub><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mstyle><mi>Θ</mi></mstyle><mo>)</mo></mrow><mo>·</mo><mi>∇</mi><msub><mi>u</mi><mi>t</mi></msub><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></math></span></span></span>which in the case <span><math><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow></math></span> and with <em>γ</em> ≡ Γ as well as <em>f</em> ≡ <em>F</em> reduces to the classical model for the evolution of displacement and temperatures in thermoviscoelasticity. Unlike in previous related studies, the focus here is on situations in which besides <em>f</em> and <em>F</em>, also the core ingredients <em>γ</em> and Γ may depend on the temperature variable Θ. Firstly, a statement on local existence of classical solutions is derived for arbitrary <em>a</em> > 0, <em>D</em> > 0 as well as 0 < <em>γ</em> ∈ <em>C</em><sup>2</sup>([0, ∞)) and 0 ≤ Γ ∈ <em>C</em><sup>1</sup>([0, ∞)), for functions <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>;</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>F</mi><mo>∈</mo><msup><mi>C</mi><mn>1</mn></msup><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>;</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>F</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></mrow></math></span>, and for suitably regular initial data of arbitrary size. Secondly, it is seen that under an additional assumption on smallness of <em>a, f</em>′ and <em>F</em>, as well as on the deviation of the initial data from the constant state given by <span><math><mrow><mi>u</mi><mo>=</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mstyle><mi>Θ</mi></mstyle><mo>=</mo><msub><mstyle><mi>Θ</mi></mstyle><mi>★</mi>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104580"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2025-12-23DOI: 10.1016/j.nonrwa.2025.104573
Rafael Muñoz-Sola
The aim of this paper is to study a model of electromagnetic levitation for a metallic rigid body. The model is constituted by the transient linear model of eddy currents under the hypothesis of axisymmetry, written in terms of a magnetic potential vector, coupled with an ODE which governs the vertical motion of the body. The electromagnetic model is a parabolic-elliptic PDE which parabolicity region is the position occupied by the body, which changes with time. Besides, Lorentz force appears in the RHS of the ODE. Thus, the model exhibits a coupling of geometrical nature. We establish the existence and uniqueness of solution of the coupled problem and we study its maximally defined solution. In particular, we prove that a blow-up of the velocity of the body cannot happen. Our techniques involve: a reformulation of the coupled problem as a causal differential equation, an adaptation of the theory about this kind of equations and a result of locally Lipschitz dependence of the magnetic potential vector with respect to the velocity of the body.
{"title":"Mathematical analysis of a levitation model","authors":"Rafael Muñoz-Sola","doi":"10.1016/j.nonrwa.2025.104573","DOIUrl":"10.1016/j.nonrwa.2025.104573","url":null,"abstract":"<div><div>The aim of this paper is to study a model of electromagnetic levitation for a metallic rigid body. The model is constituted by the transient linear model of eddy currents under the hypothesis of axisymmetry, written in terms of a magnetic potential vector, coupled with an ODE which governs the vertical motion of the body. The electromagnetic model is a parabolic-elliptic PDE which parabolicity region is the position occupied by the body, which changes with time. Besides, Lorentz force appears in the RHS of the ODE. Thus, the model exhibits a coupling of geometrical nature. We establish the existence and uniqueness of solution of the coupled problem and we study its maximally defined solution. In particular, we prove that a blow-up of the velocity of the body cannot happen. Our techniques involve: a reformulation of the coupled problem as a causal differential equation, an adaptation of the theory about this kind of equations and a result of locally Lipschitz dependence of the magnetic potential vector with respect to the velocity of the body.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104573"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-01-19DOI: 10.1016/j.nonrwa.2026.104603
José Paulo Carvalho dos Santos , Evandro Monteiro , Nelson Henrique Teixeira Lemes , Ana Claudia Pereira
The focus of this research is an epidemic model that examines the spread of rabies in the bovine population, with the spatial diffusion in the bat population, which serves as the vector population. The study investigates both the well-posedness and qualitative behavior of equilibrium points. The paper establishes the well-posedness of the model through Semigroup theory of sectorial operators and existence results for abstract parabolic differential equations. The research also addresses the definition of the basic reproduction number, , which acts as a threshold index point using linearization theory for reaction-diffusion equations in the disease-free equilibrium point. Additionally, the global asymptotic stability is established through the use of a Lyapunov function and energy estimates.
{"title":"An epidemic model for bovine rabies transmission by bats with spatial diffusion","authors":"José Paulo Carvalho dos Santos , Evandro Monteiro , Nelson Henrique Teixeira Lemes , Ana Claudia Pereira","doi":"10.1016/j.nonrwa.2026.104603","DOIUrl":"10.1016/j.nonrwa.2026.104603","url":null,"abstract":"<div><div>The focus of this research is an epidemic model that examines the spread of rabies in the bovine population, with the spatial diffusion in the bat population, which serves as the vector population. The study investigates both the well-posedness and qualitative behavior of equilibrium points. The paper establishes the well-posedness of the model through Semigroup theory of sectorial operators and existence results for abstract parabolic differential equations. The research also addresses the definition of the basic reproduction number, <span><math><msub><mi>R</mi><mn>0</mn></msub></math></span>, which acts as a threshold index point using linearization theory for reaction-diffusion equations in the disease-free equilibrium point. Additionally, the global asymptotic stability is established through the use of a Lyapunov function and energy estimates.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104603"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146037633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-10-01Epub Date: 2026-01-31DOI: 10.1016/j.nonrwa.2026.104613
Boubakr Lamouri , Ahmed Boudaoui , Salih Djilali
We investigate a nonlocal SIS epidemic model that incorporates distinct mobility patterns for susceptible and infected individuals, together with a logistic growth. The model includes distinct nonlocal diffusion kernels, denoted by J1(x) and J2(x), which represent different mobility strategies of the susceptible and infected populations, respectively. This formulation enhances the biological realism of the model by allowing greater flexibility in the representation of individual movement behaviors. Consequently, it introduces additional mathematical challenges in the analysis while providing a more accurate modelling for studying the spatial spread of infectious diseases. We establish the well-posedness, positivity, and uniform boundedness of solutions, and prove the existence of a global attractor. The basic reproduction number is derived, and persistence theory is used to show the existence of an endemic steady state when . We further analyze the asymptotic profiles of the endemic steady states under extreme diffusion limits, highlighting the impact of mobility on disease persistence.
{"title":"Effect of diffusion rates on a nonlocal SIS model with distinct dispersal kernels and logistic source","authors":"Boubakr Lamouri , Ahmed Boudaoui , Salih Djilali","doi":"10.1016/j.nonrwa.2026.104613","DOIUrl":"10.1016/j.nonrwa.2026.104613","url":null,"abstract":"<div><div>We investigate a nonlocal SIS epidemic model that incorporates distinct mobility patterns for susceptible and infected individuals, together with a logistic growth. The model includes distinct nonlocal diffusion kernels, denoted by <strong>J</strong><sub>1</sub>(<em>x</em>) and <strong>J</strong><sub>2</sub>(<em>x</em>), which represent different mobility strategies of the susceptible and infected populations, respectively. This formulation enhances the biological realism of the model by allowing greater flexibility in the representation of individual movement behaviors. Consequently, it introduces additional mathematical challenges in the analysis while providing a more accurate modelling for studying the spatial spread of infectious diseases. We establish the well-posedness, positivity, and uniform boundedness of solutions, and prove the existence of a global attractor. The basic reproduction number <span><math><msub><mi>R</mi><mn>0</mn></msub></math></span> is derived, and persistence theory is used to show the existence of an endemic steady state when <span><math><mrow><msub><mi>R</mi><mn>0</mn></msub><mo>></mo><mn>1</mn></mrow></math></span>. We further analyze the asymptotic profiles of the endemic steady states under extreme diffusion limits, highlighting the impact of mobility on disease persistence.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104613"},"PeriodicalIF":1.8,"publicationDate":"2026-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146188908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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