Pub Date : 2024-10-05DOI: 10.1016/j.nonrwa.2024.104228
Chunqiu Li, Jintao Wang
In this article we study the dynamic bifurcation of nonautonomous evolution equations by using cohomology methods. First, we construct a homotopy equivalence relation between the nonautonomous system and a product flow. Then, we slightly extend some continuation theorems on bifurcations for autonomous equations, and prove some new cohomology consequences on the reduced singular groups. Based on this homotopy equivalence relation and these conclusions, we establish some typical results on the dynamic bifurcation from infinity of the abstract nonautonomous evolution equation. Finally, we consider the parabolic equation associated with the Dirichlet boundary condition, where satisfies the appropriate Landesman–Lazer type condition. Some new results on the dynamical behaviors of this equation near resonance of the equation are derived.
{"title":"Dynamic bifurcation of nonautonomous evolution equations under Landesman–Lazer condition with cohomology methods","authors":"Chunqiu Li, Jintao Wang","doi":"10.1016/j.nonrwa.2024.104228","DOIUrl":"10.1016/j.nonrwa.2024.104228","url":null,"abstract":"<div><div>In this article we study the dynamic bifurcation of nonautonomous evolution equations by using cohomology methods. First, we construct a homotopy equivalence relation between the nonautonomous system and a product flow. Then, we slightly extend some continuation theorems on bifurcations for autonomous equations, and prove some new cohomology consequences on the reduced singular groups. Based on this homotopy equivalence relation and these conclusions, we establish some typical results on the dynamic bifurcation from infinity of the abstract nonautonomous evolution equation. Finally, we consider the parabolic equation <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>λ</mi><mi>u</mi><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> associated with the Dirichlet boundary condition, where <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> satisfies the appropriate Landesman–Lazer type condition. Some new results on the dynamical behaviors of this equation near resonance of the equation are derived.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"82 ","pages":"Article 104228"},"PeriodicalIF":1.8,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423045","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}
We consider integral functionals with slow growth and explicit dependence on of the Lagrangian; this includes many relevant examples as, for instance, in elastoplastic torsion problems or in image restoration problems. Our aim is to prove that the local minimizers are locally Lipschitz continuous. The proof makes use of recent results concerning the Bounded Slope Conditions.
我们考虑的是增长缓慢且明确依赖于拉格朗日 u 的积分函数；这包括许多相关的例子，例如弹塑性扭转问题或图像复原问题。我们的目的是证明局部最小值是局部 Lipschitz 连续的。该证明利用了有关有界斜率条件的最新成果。
{"title":"Local Lipschitz continuity for energy integrals with slow growth and lower order terms","authors":"Michela Eleuteri , Stefania Perrotta , Giulia Treu","doi":"10.1016/j.nonrwa.2024.104224","DOIUrl":"10.1016/j.nonrwa.2024.104224","url":null,"abstract":"<div><div>We consider integral functionals with slow growth and explicit dependence on <span><math><mi>u</mi></math></span> of the Lagrangian; this includes many relevant examples as, for instance, in elastoplastic torsion problems or in image restoration problems. Our aim is to prove that the local minimizers are locally Lipschitz continuous. The proof makes use of recent results concerning the Bounded Slope Conditions.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"82 ","pages":"Article 104224"},"PeriodicalIF":1.8,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.nonrwa.2024.104226
Chuangxia Huang , Xiaodan Ding
In this paper, the famous Bernfeld–Haddock conjecture is generalized to a broader form combining with a class of non-autonomous delay differential equations. With the help of differential inequality technique and Dini derivative theory, it is proved that each solution of the addressed equations has boundedness and tends to a constant without requiring the delay feedback function to be strictly increasing, which greatly refines and extends the corresponding results in the existing literature. In particular, an explanatory example is performed to substantiate the obtained analytical findings.
{"title":"New generalization of non-autonomous Bernfeld–Haddock conjecture and its proof","authors":"Chuangxia Huang , Xiaodan Ding","doi":"10.1016/j.nonrwa.2024.104226","DOIUrl":"10.1016/j.nonrwa.2024.104226","url":null,"abstract":"<div><div>In this paper, the famous Bernfeld–Haddock conjecture is generalized to a broader form combining with a class of non-autonomous delay differential equations. With the help of differential inequality technique and Dini derivative theory, it is proved that each solution of the addressed equations has boundedness and tends to a constant without requiring the delay feedback function to be strictly increasing, which greatly refines and extends the corresponding results in the existing literature. In particular, an explanatory example is performed to substantiate the obtained analytical findings.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"82 ","pages":"Article 104226"},"PeriodicalIF":1.8,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142359400","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 : 2024-09-27DOI: 10.1016/j.nonrwa.2024.104229
Shiping Lu , Xingchen Yu , Zhuomo An
<div><div>In this paper, we study the oscillations of an idealized mass–spring model of micro-electro-mechanical system (MEMS) with squeeze film damping. The model consists of two parallel electrodes separated by a gap <span><math><mi>d</mi></math></span>: one of them is fixed, and another one is movable and attached to a linear spring with stiffness coefficient <span><math><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow></math></span>. The oscillation, under the influence of AC–DC voltage <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>d</mi><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>a</mi><mi>c</mi></mrow></msub><mo>cos</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mi>T</mi></mrow></mfrac><mi>t</mi></mrow></math></span>, is ruled by the following singular differential equation <span><span><span><math><mrow><mi>m</mi><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo>+</mo><mrow><mo>[</mo><mrow><mfrac><mrow><mi>A</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>d</mi><mo>−</mo><mi>y</mi></mrow></mfrac></mrow><mo>]</mo></mrow><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>+</mo><mi>k</mi><mi>y</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub><mi>A</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mfrac><mrow><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>.</mo></mrow></math></span></span></span>Here, <span><math><mi>y</mi></math></span> is the vertical displacement of the moving plate (<span><math><mi>y</mi></math></span> is always assumed to be less than <span><math><mi>d</mi></math></span>), <span><math><mrow><mi>m</mi><mo>></mo><mn>0</mn></mrow></math></span> is its mass, <span><math><mrow><mi>A</mi><mo>></mo><mn>0</mn></mrow></math></span> is the electrode area, and <span><math><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> is the absolute dielectric constant of vacuum. Taking <span><math><mi>d</mi></math></span> as the parameter, we show the existence of saddle–node bifurcation of <span><math><mi>T</mi></math></span>-periodic solutions to the equation in the parameter space. This answers, from certain point of view, the open problem proposed by Torres in his monograph, see Torres (2015, Open Problem 2.1, p. 18). Further, we prove that the equation has exactly two classes of <span><math><mi>T</mi></math></span>-periodic solutions: as <span><math><mi>d</mi></math></span> tends to <span><math><mrow><mo>+</mo><mi>∞</mi></mrow></math></span>, one of them uniformly tends t
本文研究了具有挤压膜阻尼的微机电系统（MEMS）理想化质量弹簧模型的振荡。该模型由两个平行电极组成，两电极之间有间隙 d：其中一个固定，另一个可移动，并连接到刚度系数为 k>0 的线性弹簧上。在交直流电压 V(t)=vdc+vaccos2πTt 的影响下，振荡受以下奇异微分方程 my′′+[A(d-y)3+Ad-y]y′+ky=θ0A2V2(t)(d-y)2 的支配。这里，y 是移动板的垂直位移（y 始终假定小于 d），m>0 是移动板的质量，A>0 是电极面积，θ0>0 是真空的绝对介电常数。以 d 为参数，我们证明了方程在参数空间中存在 T 周期解的鞍节点分岔。这从某种角度回答了托雷斯在其专著中提出的开放问题，见托雷斯（2015，开放问题 2.1，第 18 页）。此外，我们还证明了方程正好有两类 T 周期解：当 d 趋于 +∞ 时，其中一类以 d 的速率均匀地趋于 +∞，而第二类的最小值趋于或越过 0。
{"title":"Bifurcation and dynamics of periodic solutions of MEMS model with squeeze film damping","authors":"Shiping Lu , Xingchen Yu , Zhuomo An","doi":"10.1016/j.nonrwa.2024.104229","DOIUrl":"10.1016/j.nonrwa.2024.104229","url":null,"abstract":"<div><div>In this paper, we study the oscillations of an idealized mass–spring model of micro-electro-mechanical system (MEMS) with squeeze film damping. The model consists of two parallel electrodes separated by a gap <span><math><mi>d</mi></math></span>: one of them is fixed, and another one is movable and attached to a linear spring with stiffness coefficient <span><math><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow></math></span>. The oscillation, under the influence of AC–DC voltage <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>d</mi><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>a</mi><mi>c</mi></mrow></msub><mo>cos</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mi>T</mi></mrow></mfrac><mi>t</mi></mrow></math></span>, is ruled by the following singular differential equation <span><span><span><math><mrow><mi>m</mi><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo>+</mo><mrow><mo>[</mo><mrow><mfrac><mrow><mi>A</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mi>d</mi><mo>−</mo><mi>y</mi></mrow></mfrac></mrow><mo>]</mo></mrow><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>+</mo><mi>k</mi><mi>y</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub><mi>A</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mfrac><mrow><msup><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>.</mo></mrow></math></span></span></span>Here, <span><math><mi>y</mi></math></span> is the vertical displacement of the moving plate (<span><math><mi>y</mi></math></span> is always assumed to be less than <span><math><mi>d</mi></math></span>), <span><math><mrow><mi>m</mi><mo>></mo><mn>0</mn></mrow></math></span> is its mass, <span><math><mrow><mi>A</mi><mo>></mo><mn>0</mn></mrow></math></span> is the electrode area, and <span><math><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> is the absolute dielectric constant of vacuum. Taking <span><math><mi>d</mi></math></span> as the parameter, we show the existence of saddle–node bifurcation of <span><math><mi>T</mi></math></span>-periodic solutions to the equation in the parameter space. This answers, from certain point of view, the open problem proposed by Torres in his monograph, see Torres (2015, Open Problem 2.1, p. 18). Further, we prove that the equation has exactly two classes of <span><math><mi>T</mi></math></span>-periodic solutions: as <span><math><mi>d</mi></math></span> tends to <span><math><mrow><mo>+</mo><mi>∞</mi></mrow></math></span>, one of them uniformly tends t","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104229"},"PeriodicalIF":1.8,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328262","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 : 2024-09-25DOI: 10.1016/j.nonrwa.2024.104227
J.A. Cardoso , J.C. de Albuquerque , J. Carvalho , G.M. Figueiredo
In this work, we study existence of positive solutions to a class of -equations in the zero mass case in . We establish a weighted Sobolev embedding and we introduce a new Trudinger–Moser type inequality. Moreover, since we work on a suitable radial Sobolev space, we prove an appropriate version of the well-known Symmetric Criticality Principle by Palais. Finally, we study regularity of solutions applying Moser iteration scheme.
{"title":"On a planar equation involving (2,q)-Laplacian with zero mass and Trudinger–Moser nonlinearity","authors":"J.A. Cardoso , J.C. de Albuquerque , J. Carvalho , G.M. Figueiredo","doi":"10.1016/j.nonrwa.2024.104227","DOIUrl":"10.1016/j.nonrwa.2024.104227","url":null,"abstract":"<div><div>In this work, we study existence of positive solutions to a class of <span><math><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span>-equations in the zero mass case in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. We establish a weighted Sobolev embedding and we introduce a new Trudinger–Moser type inequality. Moreover, since we work on a suitable radial Sobolev space, we prove an appropriate version of the well-known Symmetric Criticality Principle by Palais. Finally, we study regularity of solutions applying Moser iteration scheme.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104227"},"PeriodicalIF":1.8,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142318795","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 : 2024-09-23DOI: 10.1016/j.nonrwa.2024.104225
Nikolaos S. Papageorgiou , Dongdong Qin , Vicenţiu D. Rădulescu
We consider a parametric non-autonomous -equation with a singular term and competing nonlinearities, a parametric concave term and a Carathéodory perturbation. We consider the cases where the perturbation is -linear and where it is -superlinear (but without the use of the Ambrosetti–Rabinowitz condition). We prove an existence and multiplicity result which is global in the parameter (a bifurcation type result). Also, we show the existence of a smallest positive solution and show that it is strictly increasing as a function of the parameter. Finally, we examine the set of positive solutions as a function of the parameter (solution multifunction). First, we show that the solution set is compact in and then we show that the solution multifunction is Vietoris continuous and also Hausdorff continuous as a multifunction of the parameter.
{"title":"Singular non-autonomous (p,q)-equations with competing nonlinearities","authors":"Nikolaos S. Papageorgiou , Dongdong Qin , Vicenţiu D. Rădulescu","doi":"10.1016/j.nonrwa.2024.104225","DOIUrl":"10.1016/j.nonrwa.2024.104225","url":null,"abstract":"<div><div>We consider a parametric non-autonomous <span><math><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span>-equation with a singular term and competing nonlinearities, a parametric concave term and a Carathéodory perturbation. We consider the cases where the perturbation is <span><math><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-linear and where it is <span><math><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-superlinear (but without the use of the Ambrosetti–Rabinowitz condition). We prove an existence and multiplicity result which is global in the parameter <span><math><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></math></span> (a bifurcation type result). Also, we show the existence of a smallest positive solution and show that it is strictly increasing as a function of the parameter. Finally, we examine the set of positive solutions as a function of the parameter (solution multifunction). First, we show that the solution set is compact in <span><math><mrow><msubsup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span> and then we show that the solution multifunction is Vietoris continuous and also Hausdorff continuous as a multifunction of the parameter.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104225"},"PeriodicalIF":1.8,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311429","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 : 2024-09-23DOI: 10.1016/j.nonrwa.2024.104219
Jihoon Lee , Thanhnguyen Nguyen
In this paper we study the stability and continuity of inertial manifolds for semilinear parabolic equations. More precisely, we prove the continuity of inertial manifolds and the Gromov–Hausdorff stability of dynamical systems on inertial manifolds for reaction diffusion equations under Lipschitz perturbations of the domain and equation, using a nontrivial generalization of ODE approach discussed in Romanov (1994).
{"title":"Stability of inertial manifolds for semilinear parabolic equations under Lipschitz perturbations","authors":"Jihoon Lee , Thanhnguyen Nguyen","doi":"10.1016/j.nonrwa.2024.104219","DOIUrl":"10.1016/j.nonrwa.2024.104219","url":null,"abstract":"<div><div>In this paper we study the stability and continuity of inertial manifolds for semilinear parabolic equations. More precisely, we prove the continuity of inertial manifolds and the Gromov–Hausdorff stability of dynamical systems on inertial manifolds for reaction diffusion equations under Lipschitz perturbations of the domain and equation, using a nontrivial generalization of ODE approach discussed in Romanov (1994).</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104219"},"PeriodicalIF":1.8,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311428","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 : 2024-09-19DOI: 10.1016/j.nonrwa.2024.104223
Minzhi Wei , Feiting Fan , Xingwu Chen
In this paper, we consider a Boussinesq equation containing weak backward diffusion, delay in the convection term, dissipation and Marangoni effect. By applying geometric singular perturbation theory, a locally invariant manifold diffeomorphic to the critical manifold is established. For Boussinesq equation with delay and weak backward diffusion, the monotonicity of ratio of Abelian integrals is analyzed by utilizing the Picard–Fuchs equation. The conditions on existence of a unique periodic wave and solitary waves are obtained as well as the bound of wave speed. For Boussinesq equation with weak backward diffusion, dissipation and Marangoni effect, the corresponding Melnikov function containing three generic elements is given. The parametric conditions on existence of a unique and two periodic waves are derived respectively. Furthermore, the existence of a unique solitary wave is proved under some parametric conditions.
{"title":"Existence of periodic and solitary waves of a Boussinesq equation under perturbations","authors":"Minzhi Wei , Feiting Fan , Xingwu Chen","doi":"10.1016/j.nonrwa.2024.104223","DOIUrl":"10.1016/j.nonrwa.2024.104223","url":null,"abstract":"<div><p>In this paper, we consider a Boussinesq equation containing weak backward diffusion, delay in the convection term, dissipation and Marangoni effect. By applying geometric singular perturbation theory, a locally invariant manifold diffeomorphic to the critical manifold is established. For Boussinesq equation with delay and weak backward diffusion, the monotonicity of ratio of Abelian integrals is analyzed by utilizing the Picard–Fuchs equation. The conditions on existence of a unique periodic wave and solitary waves are obtained as well as the bound of wave speed. For Boussinesq equation with weak backward diffusion, dissipation and Marangoni effect, the corresponding Melnikov function containing three generic elements is given. The parametric conditions on existence of a unique and two periodic waves are derived respectively. Furthermore, the existence of a unique solitary wave is proved under some parametric conditions.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104223"},"PeriodicalIF":1.8,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274617","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 : 2024-09-17DOI: 10.1016/j.nonrwa.2024.104221
Luís Barreira , Matheus G.C. Cunha , Claudia Valls
We show that the hyperbolicity of a linear delay-difference equation with infinite delay, expressed in terms of the existence of an exponential dichotomy, can be completely characterized by the hyperbolicity of a linear cocycle obtained from the solutions of the equation. As an application of this characterization, we obtain several consequences: the extension of hyperbolicity to all equations in the invariant hull; the robustness of the existence of hyperbolicity for all equations in this hull under sufficiently small linear perturbations; the equality of all spectra in the invariant hull; and a characterization of hyperbolicity for all equations in the invariant hull in terms of an admissibility property taking bounded perturbations to bounded solutions.
{"title":"Cocycles for equations with infinite delay and hyperbolicity","authors":"Luís Barreira , Matheus G.C. Cunha , Claudia Valls","doi":"10.1016/j.nonrwa.2024.104221","DOIUrl":"10.1016/j.nonrwa.2024.104221","url":null,"abstract":"<div><p>We show that the hyperbolicity of a linear delay-difference equation with <em>infinite delay</em>, expressed in terms of the existence of an exponential dichotomy, can be completely characterized by the hyperbolicity of a linear cocycle obtained from the solutions of the equation. As an application of this characterization, we obtain several consequences: the extension of hyperbolicity to all equations in the invariant hull; the robustness of the existence of hyperbolicity for all equations in this hull under sufficiently small linear perturbations; the equality of all spectra in the invariant hull; and a characterization of hyperbolicity for all equations in the invariant hull in terms of an admissibility property taking bounded perturbations to bounded solutions.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104221"},"PeriodicalIF":1.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142241428","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 : 2024-09-17DOI: 10.1016/j.nonrwa.2024.104222
He Bao, Yaoning Jia, Qian Zhang
We consider the four-component chemotaxis-Navier–Stokes system in : Utilizing the Fourier localization technique alongside the inherent structure of the equations, we achieve global well-posedness for a class of rough initial data in the context of the 2D incompressible four-component chemotaxis-Navier–Stokes equations with gradient-dependent flux limitation for .
我们考虑 R2 中的四分量化合-纳维尔-斯托克斯系统：nt+u⋅∇n=Δn-∇⋅(nf(|∇c|2)∇c)-nm,ct+u⋅∇c=Δc-c+m,mt+u⋅∇m=Δm-nm,ut+(u⋅∇)u+∇P=Δu+(n+m)∇j,∇⋅u=0。利用傅立叶局部化技术和方程的固有结构，我们在二维不可压缩的四分量趋化-纳维尔-斯托克斯方程的背景下，对一类粗糙初始数据实现了全局良好求解，该方程具有梯度依赖通量限制 f(ζ)=Kf⋅(1+ζ)-α2 for α>0。
{"title":"On the global well-posedness for the incompressible four-component chemotaxis-Navier–Stokes equations with gradient-dependent flux limitation in R2","authors":"He Bao, Yaoning Jia, Qian Zhang","doi":"10.1016/j.nonrwa.2024.104222","DOIUrl":"10.1016/j.nonrwa.2024.104222","url":null,"abstract":"<div><p>We consider the four-component chemotaxis-Navier–Stokes system in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>: <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>u</mi><mi>⋅</mi><mo>∇</mo><mi>n</mi><mo>=</mo><mi>Δ</mi><mi>n</mi><mo>−</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>n</mi><mi>f</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>c</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>∇</mo><mi>c</mi><mo>)</mo></mrow><mo>−</mo><mi>n</mi><mi>m</mi><mo>,</mo><mspace></mspace></mtd></mtr><mtr><mtd><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>u</mi><mi>⋅</mi><mo>∇</mo><mi>c</mi><mo>=</mo><mi>Δ</mi><mi>c</mi><mo>−</mo><mi>c</mi><mo>+</mo><mi>m</mi><mo>,</mo><mspace></mspace></mtd></mtr><mtr><mtd><msub><mrow><mi>m</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>u</mi><mi>⋅</mi><mo>∇</mo><mi>m</mi><mo>=</mo><mi>Δ</mi><mi>m</mi><mo>−</mo><mi>n</mi><mi>m</mi><mo>,</mo><mspace></mspace></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mrow><mo>(</mo><mi>u</mi><mi>⋅</mi><mo>∇</mo><mo>)</mo></mrow><mi>u</mi><mo>+</mo><mo>∇</mo><mi>P</mi><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo></mrow><mo>∇</mo><mi>ϕ</mi><mo>,</mo><mspace></mspace></mtd></mtr><mtr><mtd><mo>∇</mo><mi>⋅</mi><mi>u</mi><mo>=</mo><mn>0</mn><mo>.</mo><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>Utilizing the Fourier localization technique alongside the inherent structure of the equations, we achieve global well-posedness for a class of rough initial data in the context of the 2D incompressible four-component chemotaxis-Navier–Stokes equations with gradient-dependent flux limitation <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>ζ</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>f</mi></mrow></msub><mi>⋅</mi><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ζ</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math></span> for <span><math><mrow><mi>α</mi><mo>></mo><mn>0</mn></mrow></math></span>.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104222"},"PeriodicalIF":1.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142241426","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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