Pub Date : 2024-09-27DOI: 10.1016/j.nonrwa.2024.104229
<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":"","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":null,"pages":null},"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
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":"","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":null,"pages":null},"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
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":"","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":null,"pages":null},"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
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":"","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":null,"pages":null},"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
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":"","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":null,"pages":null},"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
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":"","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":null,"pages":null},"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
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":"","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":null,"pages":null},"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}
Pub Date : 2024-09-17DOI: 10.1016/j.nonrwa.2024.104220
In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.
{"title":"Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity","authors":"","doi":"10.1016/j.nonrwa.2024.104220","DOIUrl":"10.1016/j.nonrwa.2024.104220","url":null,"abstract":"<div><p>In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142241427","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 : 2024-09-16DOI: 10.1016/j.nonrwa.2024.104207
We study the low Mach number limit of the compressible magnetohydrodynamic equations in a bounded domain with ill-prepared initial data. The velocity field satisfies the Navier-slip boundary conditions and the magnetic field satisfies the perfectly conducting boundary conditions. By performing energy estimate in the conormal Sobolev space and proving the maximum principle to the equations satisfied by , we overcome the difficulties caused by the simultaneous occurrence of fast oscillation and boundary layer. As a consequence, the uniform existence and the convergence of solutions are obtained.
{"title":"Incompressible limit of the compressible magnetohydrodynamic equations with ill-prepared data in a perfectly conducting container","authors":"","doi":"10.1016/j.nonrwa.2024.104207","DOIUrl":"10.1016/j.nonrwa.2024.104207","url":null,"abstract":"<div><p>We study the low Mach number limit of the compressible magnetohydrodynamic equations in a bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span> with ill-prepared initial data. The velocity field satisfies the Navier-slip boundary conditions and the magnetic field satisfies the perfectly conducting boundary conditions. By performing energy estimate in the conormal Sobolev space and proving the maximum principle to the equations satisfied by <span><math><mrow><mo>(</mo><mo>∇</mo><mo>×</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>,</mo><mo>∇</mo><mo>×</mo><msup><mrow><mi>B</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>)</mo></mrow></math></span>, we overcome the difficulties caused by the simultaneous occurrence of fast oscillation and boundary layer. As a consequence, the uniform existence and the convergence of solutions are obtained.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142241425","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 : 2024-09-14DOI: 10.1016/j.nonrwa.2024.104217
In this paper, we shall investigate an initial–boundary value problem of a generalized Swift–Hohenberg model subject to homogeneous Dirichlet boundary conditions in two spatial dimensions. The model consists of a nonlinear term of the form in the free energy functional, which is used to model the stability of fronts between hexagons and squares in pinning effect. We first prove the global-in-time existence and uniqueness of weak solutions to this initial–boundary value problem in the case with the parameter , where we employ the energy method and make use of various techniques to derive delicate a priori estimates. At the end, a few numerical experiments of the model are also performed to study the competition between hexagons and squares.
{"title":"Existence of global weak solutions and simulations to a Dirichlet problem for a generalized Swift–Hohenberg equation","authors":"","doi":"10.1016/j.nonrwa.2024.104217","DOIUrl":"10.1016/j.nonrwa.2024.104217","url":null,"abstract":"<div><p>In this paper, we shall investigate an initial–boundary value problem of a generalized Swift–Hohenberg model subject to homogeneous Dirichlet boundary conditions in two spatial dimensions. The model consists of a nonlinear term of the form <span><math><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>ψ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> in the free energy functional, which is used to model the stability of fronts between hexagons and squares in pinning effect. We first prove the global-in-time existence and uniqueness of weak solutions to this initial–boundary value problem in the case with the parameter <span><math><mrow><mi>β</mi><mo><</mo><mn>0</mn></mrow></math></span>, where we employ the energy method and make use of various techniques to derive delicate <em>a priori</em> estimates. At the end, a few numerical experiments of the model are also performed to study the competition between hexagons and squares.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142232745","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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