Pub Date : 2025-04-26DOI: 10.1016/j.na.2025.113817
L. Desvillettes , L. Fiorentino , T. Mautone
We consider a system of three reaction–diffusion equations modeling the interaction between a prey species and two predators species including functional responses of Holly type-II and Leslie–Gower type. We propose a reaction–diffusion model with five equations with simpler functional responses which, in the fast reaction limit, allows to recover the zero-th order terms of the initially considered system. The diffusive part of the initial equations is however modified and cross diffusion terms pop up. We first study the equilibria of this new system and show that no Turing instability appears. We then rigorously prove a partial result of convergence for the fast reaction limit (in 1D and 2D).
{"title":"Fast reaction limit for a Leslie–Gower model including preys, meso-predators and top-predators","authors":"L. Desvillettes , L. Fiorentino , T. Mautone","doi":"10.1016/j.na.2025.113817","DOIUrl":"10.1016/j.na.2025.113817","url":null,"abstract":"<div><div>We consider a system of three reaction–diffusion equations modeling the interaction between a prey species and two predators species including functional responses of Holly type-II and Leslie–Gower type. We propose a reaction–diffusion model with five equations with simpler functional responses which, in the fast reaction limit, allows to recover the zero-th order terms of the initially considered system. The diffusive part of the initial equations is however modified and cross diffusion terms pop up. We first study the equilibria of this new system and show that no Turing instability appears. We then rigorously prove a partial result of convergence for the fast reaction limit (in 1D and 2D).</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"258 ","pages":"Article 113817"},"PeriodicalIF":1.3,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143874392","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 : 2025-04-24DOI: 10.1016/j.na.2025.113824
Jintao Li , Lu Zhu
In this paper, we are concerned with the shock formation and construction for -system when the initial data are degenerate with finite or infinite orders (see (1.6) or (1.7) below). Since the solution we consider is a simple wave before shock formation, then with the help of the construction of shock solutions for the corresponding scalar equation, we establish the detailed estimates of the approximate solutions and then prove the convergence of the approximate solutions. Thus a weak entropy solution and a shock curve starting from the blowup point are constructed under two types of degenerate conditions. Meanwhile, some precise descriptions on the behaviors of the solutions near the blowup point are given.
{"title":"Formation and construction of shock for p-system under degenerate conditions of finite or infinite orders","authors":"Jintao Li , Lu Zhu","doi":"10.1016/j.na.2025.113824","DOIUrl":"10.1016/j.na.2025.113824","url":null,"abstract":"<div><div>In this paper, we are concerned with the shock formation and construction for <span><math><mi>p</mi></math></span>-system when the initial data are degenerate with finite or infinite orders (see <span><span>(1.6)</span></span> or <span><span>(1.7)</span></span> below). Since the solution we consider is a simple wave before shock formation, then with the help of the construction of shock solutions for the corresponding scalar equation, we establish the detailed estimates of the approximate solutions and then prove the convergence of the approximate solutions. Thus a weak entropy solution and a shock curve starting from the blowup point are constructed under two types of degenerate conditions. Meanwhile, some precise descriptions on the behaviors of the solutions near the blowup point are given.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"258 ","pages":"Article 113824"},"PeriodicalIF":1.3,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868927","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 : 2025-04-24DOI: 10.1016/j.na.2025.113825
Manli Cheng , Lan Tang
In this work, we mainly consider the chord Minkowski problem and the existence results of solutions to this problem have been established by the method of flow governed by parabolic equations for the two cases: (1) and ; (2) and .
{"title":"A curvature flow approach to the Lp chord Minkowski problem","authors":"Manli Cheng , Lan Tang","doi":"10.1016/j.na.2025.113825","DOIUrl":"10.1016/j.na.2025.113825","url":null,"abstract":"<div><div>In this work, we mainly consider the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> chord Minkowski problem and the existence results of solutions to this problem have been established by the method of flow governed by parabolic equations for the two cases: (1) <span><math><mrow><mn>0</mn><mo><</mo><mi>p</mi><mo>≤</mo><mi>n</mi><mo>+</mo><mi>q</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>q</mi><mo>></mo><mn>2</mn></mrow></math></span>; (2) <span><math><mrow><mi>p</mi><mo>></mo><mi>n</mi></mrow></math></span> and <span><math><mrow><mi>q</mi><mo>></mo><mn>2</mn></mrow></math></span> .</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"258 ","pages":"Article 113825"},"PeriodicalIF":1.3,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869049","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 : 2025-04-19DOI: 10.1016/j.na.2025.113815
Haijing Zhao, Xuechun Yang, Baode Li
We introduce a new class of weighted local approximate atoms including classical weighted local atoms. Then we further obtain the weighted local approximate atomic decompositions of weighted local Hardy spaces with and weight . As an application, we prove the boundedness of inhomogeneous Calderón–Zygmund operators on via weighted local approximate atoms and molecules.
{"title":"New atomic decompositions of weighted local Hardy spaces","authors":"Haijing Zhao, Xuechun Yang, Baode Li","doi":"10.1016/j.na.2025.113815","DOIUrl":"10.1016/j.na.2025.113815","url":null,"abstract":"<div><div>We introduce a new class of weighted local approximate atoms including classical weighted local atoms. Then we further obtain the weighted local approximate atomic decompositions of weighted local Hardy spaces <span><math><mrow><msubsup><mrow><mi>h</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mn>0</mn><mo><</mo><mi>p</mi><mo>≤</mo><mn>1</mn></mrow></math></span> and weight <span><math><mrow><mi>ω</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>. As an application, we prove the boundedness of inhomogeneous Calderón–Zygmund operators on <span><math><mrow><msubsup><mrow><mi>h</mi></mrow><mrow><mi>ω</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> via weighted local approximate atoms and molecules.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"258 ","pages":"Article 113815"},"PeriodicalIF":1.3,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143850288","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 : 2025-04-19DOI: 10.1016/j.na.2025.113816
Jana Björn , Abubakar Mwasa
We study a mixed boundary value problem for the quasilinear elliptic equation in an open infinite circular half-cylinder with prescribed continuous Dirichlet data on a part of the boundary and zero conormal derivative on the rest. The equation is assumed to satisfy the standard ellipticity conditions with a parameter . We prove the existence and uniqueness of bounded weak solutions to the mixed problem and characterize the regularity of the point at infinity in terms of -capacities. For solutions with only Neumann data near the point at infinity we show that they behave in exactly one of three possible ways, similar to the alternatives in the Phragmén–Lindelöf principle.
{"title":"Behaviour at infinity for solutions of a mixed nonlinear elliptic boundary value problem via inversion","authors":"Jana Björn , Abubakar Mwasa","doi":"10.1016/j.na.2025.113816","DOIUrl":"10.1016/j.na.2025.113816","url":null,"abstract":"<div><div>We study a mixed boundary value problem for the quasilinear elliptic equation <span><math><mrow><mo>div</mo><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mo>∇</mo><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> in an open infinite circular half-cylinder with prescribed continuous Dirichlet data on a part of the boundary and zero conormal derivative on the rest. The equation is assumed to satisfy the standard ellipticity conditions with a parameter <span><math><mrow><mi>p</mi><mo>></mo><mn>1</mn></mrow></math></span>. We prove the existence and uniqueness of bounded weak solutions to the mixed problem and characterize the regularity of the point at infinity in terms of <span><math><mi>p</mi></math></span>-capacities. For solutions with only Neumann data near the point at infinity we show that they behave in exactly one of three possible ways, similar to the alternatives in the Phragmén–Lindelöf principle.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"258 ","pages":"Article 113816"},"PeriodicalIF":1.3,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143848384","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 : 2025-04-18DOI: 10.1016/j.na.2025.113810
David Kinderlehrer , KiHyun Yun
Microstructural coarsening is a network of grains separated by interfaces, facets, that evolve by generalized curvature. The facets meet along curves that come together at points. Systems of this nature consist of many nonlinear PDEs with their boundary conditions. Here we explore the local in time existence for a configuration, perhaps the simplest, close to a known equilibrium configuration.
{"title":"Microstructural evolution in 3D: An existence result","authors":"David Kinderlehrer , KiHyun Yun","doi":"10.1016/j.na.2025.113810","DOIUrl":"10.1016/j.na.2025.113810","url":null,"abstract":"<div><div>Microstructural coarsening is a network of grains separated by interfaces, facets, that evolve by generalized curvature. The facets meet along curves that come together at points. Systems of this nature consist of many nonlinear PDEs with their boundary conditions. Here we explore the local in time existence for a configuration, perhaps the simplest, close to a known equilibrium configuration.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"258 ","pages":"Article 113810"},"PeriodicalIF":1.3,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143844933","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 : 2025-04-12DOI: 10.1016/j.na.2025.113814
Laura Baldelli , David Krejčiřík
The Dirichlet -Laplacian in tubes of arbitrary cross-section along infinite curves in Euclidean spaces of arbitrary dimension is investigated. First, it is shown that the gap between the lowest point of the generalised spectrum and the essential spectrum is positive whenever the cross-section is centrally symmetric and the tube is asymptotically straight, untwisted and non-trivially bent. Second, a Hardy-type inequality is derived for unbent and non-trivially twisted tubes.
{"title":"Curved nonlinear waveguides","authors":"Laura Baldelli , David Krejčiřík","doi":"10.1016/j.na.2025.113814","DOIUrl":"10.1016/j.na.2025.113814","url":null,"abstract":"<div><div>The Dirichlet <span><math><mi>p</mi></math></span>-Laplacian in tubes of arbitrary cross-section along infinite curves in Euclidean spaces of arbitrary dimension is investigated. First, it is shown that the gap between the lowest point of the generalised spectrum and the essential spectrum is positive whenever the cross-section is centrally symmetric and the tube is asymptotically straight, untwisted and non-trivially bent. Second, a Hardy-type inequality is derived for unbent and non-trivially twisted tubes.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"258 ","pages":"Article 113814"},"PeriodicalIF":1.3,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143820290","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 : 2025-04-12DOI: 10.1016/j.na.2025.113809
François Bouchut , Carsten Carstensen , Alexandre Ern
The Bingham fluid model for viscoplastic materials involves the minimization of a nondifferentiable functional. The regularity of the associated solution is investigated here. The simplified scalar case is considered first: the total variation minimization problem. Our main result proves for a convex domain that a right-hand side gives a solution . Homogeneous Dirichlet conditions involve an additional trace term, then implies . In the case of the inviscid vector Bingham fluid model, boundary conditions are difficult to handle, but we prove the local regularity of the solution for . The proofs rely on several generalizations of a lemma due to Brézis and on the viscous approximation. We obtain Euler–Lagrange characterizations of the solution. Homogeneous Dirichlet conditions on the viscous problem lead in the vanishing viscosity limit to relaxed boundary conditions of frictional type.
{"title":"H1 regularity of the minimizers for the inviscid total variation and Bingham fluid problems for H1 data","authors":"François Bouchut , Carsten Carstensen , Alexandre Ern","doi":"10.1016/j.na.2025.113809","DOIUrl":"10.1016/j.na.2025.113809","url":null,"abstract":"<div><div>The Bingham fluid model for viscoplastic materials involves the minimization of a nondifferentiable functional. The regularity of the associated solution is investigated here. The simplified scalar case is considered first: the total variation minimization problem. Our main result proves for a convex domain <span><math><mi>Ω</mi></math></span> that a right-hand side <span><math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> gives a solution <span><math><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>. Homogeneous Dirichlet conditions involve an additional trace term, then <span><math><mrow><mi>f</mi><mo>∈</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> implies <span><math><mrow><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>. In the case of the inviscid vector Bingham fluid model, boundary conditions are difficult to handle, but we prove the local <span><math><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mtext>loc</mtext></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mrow><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> regularity of the solution for <span><math><mrow><mi>f</mi><mo>∈</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mtext>loc</mtext></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mrow><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>. The proofs rely on several generalizations of a lemma due to Brézis and on the viscous approximation. We obtain Euler–Lagrange characterizations of the solution. Homogeneous Dirichlet conditions on the viscous problem lead in the vanishing viscosity limit to relaxed boundary conditions of frictional type.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"258 ","pages":"Article 113809"},"PeriodicalIF":1.3,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143823986","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 : 2025-04-10DOI: 10.1016/j.na.2025.113813
Veronica Felli , Benedetta Noris , Giovanni Siclari
This paper deals with quantitative spectral stability for Aharonov-Bohm operators with many colliding poles of whichever circulation. An equivalent formulation of the eigenvalue problem is derived as a system of two equations with real coefficients, coupled through prescribed jumps of the unknowns and their normal derivatives across the segments joining the poles with the collision point. Under the assumption that the sum of all circulations is not integer, the dominant term in the asymptotic expansion for eigenvalues is characterized in terms of the minimum of an energy functional associated with the configuration of poles. Estimates of the order of vanishing of the eigenvalue variation are then deduced from a blow-up analysis, yielding sharp asymptotics in some particular examples.
{"title":"On Aharonov-Bohm operators with multiple colliding poles of any circulation","authors":"Veronica Felli , Benedetta Noris , Giovanni Siclari","doi":"10.1016/j.na.2025.113813","DOIUrl":"10.1016/j.na.2025.113813","url":null,"abstract":"<div><div>This paper deals with quantitative spectral stability for Aharonov-Bohm operators with many colliding poles of whichever circulation. An equivalent formulation of the eigenvalue problem is derived as a system of two equations with real coefficients, coupled through prescribed jumps of the unknowns and their normal derivatives across the segments joining the poles with the collision point. Under the assumption that the sum of all circulations is not integer, the dominant term in the asymptotic expansion for eigenvalues is characterized in terms of the minimum of an energy functional associated with the configuration of poles. Estimates of the order of vanishing of the eigenvalue variation are then deduced from a blow-up analysis, yielding sharp asymptotics in some particular examples.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"258 ","pages":"Article 113813"},"PeriodicalIF":1.3,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808620","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}
We consider the periodic cubic–quintic nonlinear Schrödinger equation (CQNLS)on the three-dimensional torus with . As a first result, we establish the small data well-posedness of for arbitrarily given and . By adapting the crucial perturbation arguments in Zhang (2006) to the periodic setting, we also prove that is always globally well-posed in in the case .
{"title":"On well-posedness results for the cubic–quintic NLS on T3","authors":"Yongming Luo , Xueying Yu , Haitian Yue , Zehua Zhao","doi":"10.1016/j.na.2025.113806","DOIUrl":"10.1016/j.na.2025.113806","url":null,"abstract":"<div><div>We consider the periodic cubic–quintic nonlinear Schrödinger equation <span><span><span>(CQNLS)</span><span><math><mrow><mrow><mo>(</mo><mi>i</mi><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>Δ</mi><mo>)</mo></mrow><mi>u</mi><mo>=</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>4</mn></mrow></msup><mi>u</mi></mrow></math></span></span></span>on the three-dimensional torus <span><math><msup><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>R</mi><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. As a first result, we establish the small data well-posedness of for arbitrarily given <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. By adapting the crucial perturbation arguments in Zhang (2006) to the periodic setting, we also prove that is always globally well-posed in <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> in the case <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"257 ","pages":"Article 113806"},"PeriodicalIF":1.3,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143785899","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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