Pub Date : 2025-12-31DOI: 10.1016/j.na.2025.114030
Dieter Bothe, Pierre-Étienne Druet, Robert Haller
We consider elliptic transmission problems in several space dimensions near an interface which is C1,1-diffeomorphic to an axisymmetric reference interface with a singular point of cusp type. We establish the regularity of the gradient and of the Hessian in Lp spaces up to the cusp point for local weak solutions. We obtain regularity thresholds which are different according to whether the cusp is inward or outward to the subdomain, and which depend explicitly on the opening of the interface at the cusp. Our results allow for source terms in the bulk and on the interface.
{"title":"Gradient and Hessian regularity in elliptic transmission problems near a point cusp","authors":"Dieter Bothe, Pierre-Étienne Druet, Robert Haller","doi":"10.1016/j.na.2025.114030","DOIUrl":"10.1016/j.na.2025.114030","url":null,"abstract":"<div><div>We consider elliptic transmission problems in several space dimensions near an interface which is <em>C</em><sup>1,1</sup>-diffeomorphic to an axisymmetric reference interface with a singular point of cusp type. We establish the regularity of the gradient and of the Hessian in <em>L<sup>p</sup></em> spaces up to the cusp point for local weak solutions. We obtain regularity thresholds which are different according to whether the cusp is inward or outward to the subdomain, and which depend explicitly on the opening of the interface at the cusp. Our results allow for source terms in the bulk and on the interface.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114030"},"PeriodicalIF":1.3,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884533","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}
Pub Date : 2025-12-26DOI: 10.1016/j.na.2025.114044
Enrique Aguilar , Bashar Khorbatly
We consider the system of partial differential equations proposed in [1] as an alternative to the Navier-Stokes equations. These two sets of equations differ primarily in that the former incorporates diffusive terms of mass, momentum and energy. While existence of solutions to a weak version of the diffusive system is demonstrated in [1], we further reduce the diffusive differential equations and their weak counterparts using the isentropic assumption. Under specific technical assumptions, we establish a form of uniqueness known as weak-strong uniqueness for the reduced systems. This ensures that a solution to the differential equations and a solution to the weak counterpart are equivalent provided they originate from the same initial data.
{"title":"Weak-strong uniqueness in an alternative system to the isentropic Navier-Stokes equations","authors":"Enrique Aguilar , Bashar Khorbatly","doi":"10.1016/j.na.2025.114044","DOIUrl":"10.1016/j.na.2025.114044","url":null,"abstract":"<div><div>We consider the system of partial differential equations proposed in [1] as an alternative to the Navier-Stokes equations. These two sets of equations differ primarily in that the former incorporates diffusive terms of mass, momentum and energy. While existence of solutions to a weak version of the diffusive system is demonstrated in <span><span>[1]</span></span>, we further reduce the diffusive differential equations and their weak counterparts using the isentropic assumption. Under specific technical assumptions, we establish a form of uniqueness known as weak-strong uniqueness for the reduced systems. This ensures that a solution to the differential equations and a solution to the weak counterpart are equivalent provided they originate from the same initial data.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114044"},"PeriodicalIF":1.3,"publicationDate":"2025-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840534","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}
Pub Date : 2025-12-26DOI: 10.1016/j.na.2025.114045
Chen-Gang Long , Senping Luo , Wenming Zou
In this paper, we consider the minimization problem of two dimensional lattice energyWe study this minimization problem under the classical Yukawa potential with α > 0, t > 1 and . We prove the existence of a critical value such that:
•
if then the minimizer corresponds to a hexagonal lattice configuration;
•
if then no minimizer exists.
This result provide the sharp bound βc for hexagonal lattice crystallization under Yukawa potential. Furthermore, we extend the analysis to two-component lattices, where each component is centered on the other, and obtain the same critical value βc. In this case, the minimizer transitions between a rhombic-square-rectangular configuration and a scenario where no minimizer exists.
{"title":"On minima of lattice energy under Yukawa potentials","authors":"Chen-Gang Long , Senping Luo , Wenming Zou","doi":"10.1016/j.na.2025.114045","DOIUrl":"10.1016/j.na.2025.114045","url":null,"abstract":"<div><div>In this paper, we consider the minimization problem of two dimensional lattice energy<span><span><span><math><mrow><munder><mi>min</mi><mrow><mo>|</mo><mstyle><mi>Λ</mi></mstyle><mo>|</mo><mo>=</mo><mn>1</mn></mrow></munder><msub><mi>E</mi><mi>f</mi></msub><mrow><mo>(</mo><mstyle><mi>Λ</mi></mstyle><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mtext>where</mtext><mspace></mspace><msub><mi>E</mi><mi>f</mi></msub><mrow><mo>(</mo><mstyle><mi>Λ</mi></mstyle><mo>)</mo></mrow><mo>=</mo><munder><mo>∑</mo><mrow><mi>P</mi><mo>∈</mo><mstyle><mi>Λ</mi></mstyle><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></mrow></munder><msup><mrow><mi>f</mi><mo>(</mo><mo>|</mo><mi>P</mi><mo>|</mo></mrow><mn>2</mn></msup><mrow><mo>)</mo><mo>.</mo></mrow></mrow></math></span></span></span>We study this minimization problem under the classical Yukawa potential <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>=</mo><mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>α</mi><mi>π</mi><mi>r</mi></mrow></msup><mi>r</mi></mfrac><mo>−</mo><mi>β</mi><mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mi>α</mi><mi>π</mi><mi>r</mi></mrow></msup><mi>r</mi></mfrac></mrow></math></span> with <em>α</em> > 0, <em>t</em> > 1 and <span><math><mrow><mi>β</mi><mo>∈</mo><mi>R</mi></mrow></math></span>. We prove the existence of a critical value <span><math><mrow><msub><mi>β</mi><mi>c</mi></msub><mo>=</mo><mn>1</mn></mrow></math></span> such that:<ul><li><span>•</span><span><div>if <span><math><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><msub><mi>β</mi><mi>c</mi></msub><mo>]</mo><mo>,</mo></mrow></math></span> then the minimizer corresponds to a hexagonal lattice configuration;</div></span></li><li><span>•</span><span><div>if <span><math><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><msub><mi>β</mi><mi>c</mi></msub><mo>,</mo><mo>+</mo><mi>∞</mi><mo>)</mo><mo>,</mo></mrow></math></span> then no minimizer exists.</div></span></li></ul></div><div>This result provide the sharp bound <em>β<sub>c</sub></em> for hexagonal lattice crystallization under Yukawa potential. Furthermore, we extend the analysis to two-component lattices, where each component is centered on the other, and obtain the same critical value <em>β<sub>c</sub></em>. In this case, the minimizer transitions between a rhombic-square-rectangular configuration and a scenario where no minimizer exists.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114045"},"PeriodicalIF":1.3,"publicationDate":"2025-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840535","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}
Pub Date : 2025-12-23DOI: 10.1016/j.na.2025.114043
Matteo Caggio , Gabriele Sbaiz
We consider the Navier-Stokes-Fourier system for a heat conducting compressible fluid under the effects of rotation and stratification. We investigate the low Mach, Rossby and Froude number limit towards a quasi-geostrophic balance in a stratification range between the so-called low and strong stratification regimes. The limit is studied in the context of weak solutions with ill-prepared initial data.
{"title":"Between low and strong stratification regimes for rotating heat-conducting fluids","authors":"Matteo Caggio , Gabriele Sbaiz","doi":"10.1016/j.na.2025.114043","DOIUrl":"10.1016/j.na.2025.114043","url":null,"abstract":"<div><div>We consider the Navier-Stokes-Fourier system for a heat conducting compressible fluid under the effects of rotation and stratification. We investigate the low Mach, Rossby and Froude number limit towards a quasi-geostrophic balance in a stratification range between the so-called low and strong stratification regimes. The limit is studied in the context of weak solutions with ill-prepared initial data.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114043"},"PeriodicalIF":1.3,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840533","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}
Pub Date : 2025-12-22DOI: 10.1016/j.na.2025.114037
Mirella Aoun
In this paper, we consider the following class of nonlinear parabolic equations with non-homogeneous Neumann boundary conditions:where Ω is a bounded open domain of , N ≥ 2 and T > 0. Assuming that f, resp. h belong to L1(QT), resp. L1(0, T; L1(∂Ω)), while g is an element of L2(QT)N and u is a vector field verifying specific conditions, we prove the existence and uniqueness of renormalized solutions.
{"title":"Existence and uniqueness of renormalized solutions for parabolic Neumann problem with L1 data","authors":"Mirella Aoun","doi":"10.1016/j.na.2025.114037","DOIUrl":"10.1016/j.na.2025.114037","url":null,"abstract":"<div><div>In this paper, we consider the following class of nonlinear parabolic equations with non-homogeneous Neumann boundary conditions:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mfrac><mrow><mi>∂</mi><mi>θ</mi></mrow><mrow><mi>∂</mi><mi>t</mi></mrow></mfrac><mo>−</mo><mi>div</mi><mrow><mo>(</mo><mi>A</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>θ</mi><mo>)</mo><mi>∇</mi><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><mi>u</mi><mo>·</mo><mi>∇</mi><mi>θ</mi><mo>=</mo><mi>f</mi><mo>−</mo><mi>div</mi><mi>g</mi></mrow></mtd><mtd><mrow><mtext>in</mtext><mspace></mspace><mstyle><mi>Ω</mi></mstyle><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>θ</mi><mrow><mo>(</mo><mi>t</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mi>θ</mi><mn>0</mn></msub></mrow></mtd><mtd><mrow><mtext>in</mtext><mspace></mspace><mstyle><mi>Ω</mi></mstyle><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>(</mo><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow><mi>∇</mi><mi>θ</mi><mo>−</mo><mi>g</mi><mo>)</mo><mo>·</mo><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mi>h</mi></mrow></mtd><mtd><mrow><mtext>on</mtext><mspace></mspace><mi>∂</mi><mstyle><mi>Ω</mi></mstyle><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math></span></span></span>where Ω is a bounded open domain of <span><math><msup><mi>R</mi><mi>N</mi></msup></math></span>, <em>N</em> ≥ 2 and <em>T</em> > 0. Assuming that <em>f</em>, resp. <em>h</em> belong to <em>L</em><sup>1</sup>(<em>Q<sub>T</sub></em>), resp. <em>L</em><sup>1</sup>(0, <em>T</em>; <em>L</em><sup>1</sup>(∂Ω)), while <em>g</em> is an element of <em>L</em><sup>2</sup>(<em>Q<sub>T</sub></em>)<sup><em>N</em></sup> and <em>u</em> is a vector field verifying specific conditions, we prove the existence and uniqueness of renormalized solutions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114037"},"PeriodicalIF":1.3,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840532","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}
Pub Date : 2025-12-19DOI: 10.1016/j.na.2025.114039
Taiki Takeuchi
We consider the Keller–Segel system of parabolic-elliptic type with logistic growth in the whole space , where n ≥ 3, 1 < κ < 2, and l ∈ {0, 1}. We show the existence and uniqueness of local mild solutions u for initial data under the conditions n/2 < p < n, 1 ≤ q ≤ ∞, and 2p/n < κ < 2. In addition, partial smoothing effects of the mild solutions u are investigated. More precisely, we show that u satisfy the original system in a classical sense and have the property . According to the regularities of the term with the power nonlinearity, such regularities for u seem to be optimal under the general framework beyond the physical sense.
我们考虑在整个空间Rn上具有logistic增长u - |u|κ - lul的抛物-椭圆型Keller-Segel系统，其中n ≥ 3,1 <； κ <； 2，且l ∈ {0,1}。我们展示当地温和解的存在性和唯一性u初始数据∈Bp,问−2 + n / p (Rn)条件下n / 2 & lt; p & lt; n, 1 ≤ 问 ≤ ∞,和2 p / n & lt; κ & lt; 2。此外，还研究了温和溶液u的部分平滑效应。更准确地说，我们证明了u满足经典意义上的原始系统，并且具有u∈clockk +1((0,T];L∞(Rn))∩Lloc∞(（0,T];Cκ+2(Rn)）的性质。从|u|κ−l项的幂非线性规律来看，在超出物理意义的一般框架下，u的这种规律似乎是最优的。
{"title":"Partial smoothing effects of local mild solutions of the Keller–Segel system with logistic growth in Besov spaces","authors":"Taiki Takeuchi","doi":"10.1016/j.na.2025.114039","DOIUrl":"10.1016/j.na.2025.114039","url":null,"abstract":"<div><div>We consider the Keller–Segel system of parabolic-elliptic type with logistic growth <span><math><mrow><mi>u</mi><mo>−</mo><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>κ</mi><mo>−</mo><mi>l</mi></mrow></msup><msup><mi>u</mi><mi>l</mi></msup></mrow></math></span> in the whole space <span><math><msup><mi>R</mi><mi>n</mi></msup></math></span>, where <em>n</em> ≥ 3, 1 < <em>κ</em> < 2, and <em>l</em> ∈ {0, 1}. We show the existence and uniqueness of local mild solutions <em>u</em> for initial data <span><math><mrow><mi>a</mi><mo>∈</mo><msubsup><mi>B</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn><mo>+</mo><mi>n</mi><mo>/</mo><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></math></span> under the conditions <em>n</em>/2 < <em>p</em> < <em>n</em>, 1 ≤ <em>q</em> ≤ ∞, and 2<em>p</em>/<em>n</em> < <em>κ</em> < 2. In addition, partial smoothing effects of the mild solutions <em>u</em> are investigated. More precisely, we show that <em>u</em> satisfy the original system in a classical sense and have the property <span><math><mrow><mi>u</mi><mo>∈</mo><msubsup><mi>C</mi><mrow><mtext>loc</mtext></mrow><mrow><mi>κ</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>;</mo><msup><mi>L</mi><mi>∞</mi></msup><mrow><mo>(</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow><mo>)</mo></mrow><mo>∩</mo><msubsup><mi>L</mi><mrow><mtext>loc</mtext></mrow><mi>∞</mi></msubsup><mrow><mo>(</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>;</mo><msup><mi>C</mi><mrow><mi>κ</mi><mo>+</mo><mn>2</mn></mrow></msup><mrow><mo>(</mo><msup><mi>R</mi><mi>n</mi></msup><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>. According to the regularities of the term <span><math><mrow><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>κ</mi><mo>−</mo><mi>l</mi></mrow></msup><msup><mi>u</mi><mi>l</mi></msup></mrow></math></span> with the power nonlinearity, such regularities for <em>u</em> seem to be optimal under the general framework beyond the physical sense.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114039"},"PeriodicalIF":1.3,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145797903","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}
Pub Date : 2025-12-19DOI: 10.1016/j.na.2025.114042
Flank D.M. Bezerra , Vando Narciso , Senlin Yan
This paper is dedicated to the analysis of the pullback dynamics of a non-autonomous Balakrishnan-Taylor beam with a strong damping dependent on the time and linear energy of the system. In the main result we establish the existence of a pullback attractor for the evolution process generated by the weak solutions of the system. In addition, we also prove a result of upper semicontiunity of attractors with respect to functional parameters present in the damped term.
{"title":"Pullback dynamics for a class of plate equations with time-dependent energy damping","authors":"Flank D.M. Bezerra , Vando Narciso , Senlin Yan","doi":"10.1016/j.na.2025.114042","DOIUrl":"10.1016/j.na.2025.114042","url":null,"abstract":"<div><div>This paper is dedicated to the analysis of the pullback dynamics of a non-autonomous Balakrishnan-Taylor beam with a strong damping dependent on the time and linear energy of the system. In the main result we establish the existence of a pullback attractor for the evolution process generated by the weak solutions of the system. In addition, we also prove a result of upper semicontiunity of attractors with respect to functional parameters present in the damped term.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114042"},"PeriodicalIF":1.3,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145797902","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}
Pub Date : 2025-12-18DOI: 10.1016/j.na.2025.114040
Tianlan Chen , Christopher S. Goodrich
We consider a class of nonlocal elliptic PDEs, of which one model case is the steady-state Kirchhoff-type equationwhere is the unit ball in , where n ≥ 2. Under the assumption that u satisfies Dirichlet boundary datum on , we demonstrate existence of at least one positive radially symmetric solution to the PDE by means of topological fixed point theory. Our results are valid both in the low-dimensional setting (n < p) and the high-dimensional setting (n ≥ p), though the techniques required differ between the two cases. The existence arguments utilise a specialised order cone.
{"title":"Radially symmetric solutions of nonlocal elliptic equations on the unit ball","authors":"Tianlan Chen , Christopher S. Goodrich","doi":"10.1016/j.na.2025.114040","DOIUrl":"10.1016/j.na.2025.114040","url":null,"abstract":"<div><div>We consider a class of nonlocal elliptic PDEs, of which one model case is the steady-state Kirchhoff-type equation<span><span><span><math><mrow><mo>−</mo><msubsup><mrow><mi>M</mi><mo>(</mo><mo>∥</mo><mi>D</mi><mi>u</mi><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>p</mi></msup></mrow><mi>p</mi></msubsup><mo>)</mo><mstyle><mi>Δ</mi></mstyle><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>λ</mi><mi>f</mi><mo>(</mo><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow><mo>,</mo><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo><mtext>,</mtext><mspace></mspace><mi>x</mi><mo>∈</mo><msub><mi>B</mi><mn>1</mn></msub><mo>,</mo></mrow></math></span></span></span>where <span><math><msub><mi>B</mi><mn>1</mn></msub></math></span> is the unit ball in <span><math><msup><mi>R</mi><mi>n</mi></msup></math></span>, where <em>n</em> ≥ 2. Under the assumption that <em>u</em> satisfies Dirichlet boundary datum on <span><math><mrow><mi>∂</mi><msub><mi>B</mi><mn>1</mn></msub></mrow></math></span>, we demonstrate existence of at least one positive radially symmetric solution to the PDE by means of topological fixed point theory. Our results are valid both in the low-dimensional setting (<em>n</em> < <em>p</em>) and the high-dimensional setting (<em>n</em> ≥ <em>p</em>), though the techniques required differ between the two cases. The existence arguments utilise a specialised order cone.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114040"},"PeriodicalIF":1.3,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145797901","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}
Pub Date : 2025-12-16DOI: 10.1016/j.na.2025.114032
Yan Zhuang , Yanmin Niu , Daxiong Piao
In this paper, we obtain the invariant curves of quasi-periodic reversible mappings with finite s-smoothness. Since the reversible property is difficult to maintain in the process of approximating smooth functions by analytical ones, Rüssmann’s classical method in reduction of smoothness [1] cannot be directly applied since it does not preserve the reversible property. Inspired by the fact that a reversible mapping can be regarded as the Poincaré map of a reversible differential equation, we establish a new KAM theorem for a reversible differential equation which is quasi-periodic in angle variable, and then obtain the invariant curves of the reversible mapping. Beyond that, we prove some variants of invariant curve theorems for quasi-periodic reversible mappings. As an application, the boundedness of solutions for a class of semilinear oscillator is discussed by the obtained results at last.
{"title":"Invariant curves of low smooth quasi-periodic reversible mappings","authors":"Yan Zhuang , Yanmin Niu , Daxiong Piao","doi":"10.1016/j.na.2025.114032","DOIUrl":"10.1016/j.na.2025.114032","url":null,"abstract":"<div><div>In this paper, we obtain the invariant curves of quasi-periodic reversible mappings with finite s-smoothness. Since the reversible property is difficult to maintain in the process of approximating smooth functions by analytical ones, Rüssmann’s classical method in reduction of smoothness [1] cannot be directly applied since it does not preserve the reversible property. Inspired by the fact that a reversible mapping can be regarded as the Poincaré map of a reversible differential equation, we establish a new KAM theorem for a reversible differential equation which is quasi-periodic in angle variable, and then obtain the invariant curves of the reversible mapping. Beyond that, we prove some variants of invariant curve theorems for quasi-periodic reversible mappings. As an application, the boundedness of solutions for a class of semilinear oscillator is discussed by the obtained results at last.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114032"},"PeriodicalIF":1.3,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145797904","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}
Pub Date : 2025-12-16DOI: 10.1016/j.na.2025.114035
Patrick Maheux , Vittoria Pierfelice
In this paper, we revisit the blow-up criteria for the simplest parabolic-elliptic Patlak-Keller-Segel (PKS) system in the 2D Euclidean space, including a consumption term. In the supercritical mass case M > 8π, and under an additional global assumption on the second moment (or variance) of the initial data, we establish blow-up results for a broader class of initial conditions than those traditionally considered. We also derive improved upper bounds for the maximal existence time of (PKS) solutions on the plane. These time estimates are obtained through a sharp analysis of a one-parameter differential inequality governing the evolution of the second moment of the (PKS) system.
In particular, we obtain that for any n0 with finite second order moment, λn0 for λ sufficiently large provide an initial datum yielding blow up. The current blow up criterion is also compared to the available ones in the literature.
{"title":"Revisiting the blow-up criterion and the maximal existence time for solutions of the parabolic-Elliptic keller-Segel system in 2D-euclidean space","authors":"Patrick Maheux , Vittoria Pierfelice","doi":"10.1016/j.na.2025.114035","DOIUrl":"10.1016/j.na.2025.114035","url":null,"abstract":"<div><div>In this paper, we revisit the blow-up criteria for the simplest parabolic-elliptic Patlak-Keller-Segel (PKS) system in the 2D Euclidean space, including a consumption term. In the supercritical mass case <em>M</em> > 8<em>π</em>, and under an additional global assumption on the second moment (or variance) of the initial data, we establish blow-up results for a broader class of initial conditions than those traditionally considered. We also derive improved upper bounds for the maximal existence time of (PKS) solutions on the plane. These time estimates are obtained through a sharp analysis of a one-parameter differential inequality governing the evolution of the second moment of the (PKS) system.</div><div>In particular, we obtain that for any <em>n</em><sub>0</sub> with finite second order moment, <em>λn</em><sub>0</sub> for <em>λ</em> sufficiently large provide an initial datum yielding blow up. The current blow up criterion is also compared to the available ones in the literature.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"266 ","pages":"Article 114035"},"PeriodicalIF":1.3,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145797906","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}
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