Pub Date : 2024-07-08DOI: 10.1016/j.nonrwa.2024.104146
Abdelhak Bouhamed , Abella Elkabouss , Pitágoras P. de Carvalho , Hassane Bouzahir
This paper examines a nonlinear Kirchhoff plate equation, where the control acts in bilinear form within the boundary of the mentioned equation. The objective is to construct a distributed control to guide such a system from the initial state to the desired state in the final time, while minimizing a quadratic functional cost defined as the sum of the norm difference between the aforementioned state and a desired equation with an energy term. We show how to approximate the solution of the nonlinear Kirchhoff plate equation to a desired objective, indicating the existence of optimal control in specific cases. and deriving the optimally conditions for a closed convex set. Moreover, it is shown that sufficient conditions ensures the uniqueness of control optimal. Furthermore, we provide a concise numerical methodology that involves the integration of finite element and finite difference discretization methods. The approach incorporates Newton’s linearization method to assess the computational performance of the controlled problem, using the Freefem++ software.
{"title":"Boundary optimal control problem of semi-linear Kirchhoff plate equation","authors":"Abdelhak Bouhamed , Abella Elkabouss , Pitágoras P. de Carvalho , Hassane Bouzahir","doi":"10.1016/j.nonrwa.2024.104146","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104146","url":null,"abstract":"<div><p>This paper examines a nonlinear Kirchhoff plate equation, where the control acts in bilinear form within the boundary of the mentioned equation. The objective is to construct a distributed control to guide such a system from the initial state to the desired state in the final time, while minimizing a quadratic functional cost defined as the sum of the norm difference between the aforementioned state and a desired equation with an energy term. We show how to approximate the solution of the nonlinear Kirchhoff plate equation to a desired objective, indicating the existence of optimal control in specific cases. and deriving the optimally conditions for a closed convex set. Moreover, it is shown that sufficient conditions ensures the uniqueness of control optimal. Furthermore, we provide a concise numerical methodology that involves the integration of finite element and finite difference discretization methods. The approach incorporates Newton’s linearization method to assess the computational performance of the controlled problem, using the Freefem++ software.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104146"},"PeriodicalIF":1.8,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141593949","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 : 2024-07-07DOI: 10.1016/j.nonrwa.2024.104174
Liangchen Wang, Rui Huang
<div><p>This work considers the Keller–Segel consumption system <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mrow><mo>(</mo><mi>u</mi><msup><mrow><mi>v</mi></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>)</mo></mrow><mo>+</mo><mi>a</mi><mi>u</mi><mo>−</mo><mi>b</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msup><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>u</mi><mi>v</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>under homogeneous Neumann boundary conditions in a smooth bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, where the parameters <span><math><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>γ</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, the initial data <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mn>0</mn><mrow><mo>(</mo><mo>⁄</mo><mo>≡</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> in <span><math><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span> with <span><span><span><math><mrow><msub><mrow><mo>‖</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo><</mo><mo>exp</mo><mfenced><mrow><mfrac><mrow><mo>ln</mo><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><mrow><mi>α</mi></mrow></mfrac><mi>⋅</mi><mfrac><mrow><mn>8</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></mfrac></mrow></mfenced><mo>.</mo></mrow></
{"title":"Global classical solutions to a chemotaxis consumption model involving singularly signal-dependent motility and logistic source","authors":"Liangchen Wang, Rui Huang","doi":"10.1016/j.nonrwa.2024.104174","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104174","url":null,"abstract":"<div><p>This work considers the Keller–Segel consumption system <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mrow><mo>(</mo><mi>u</mi><msup><mrow><mi>v</mi></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>)</mo></mrow><mo>+</mo><mi>a</mi><mi>u</mi><mo>−</mo><mi>b</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msup><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>u</mi><mi>v</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>under homogeneous Neumann boundary conditions in a smooth bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, where the parameters <span><math><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>b</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>γ</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, the initial data <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̄</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mn>0</mn><mrow><mo>(</mo><mo>⁄</mo><mo>≡</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> in <span><math><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span> with <span><span><span><math><mrow><msub><mrow><mo>‖</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></msub><mo><</mo><mo>exp</mo><mfenced><mrow><mfrac><mrow><mo>ln</mo><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><mrow><mi>α</mi></mrow></mfrac><mi>⋅</mi><mfrac><mrow><mn>8</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></mfrac></mrow></mfenced><mo>.</mo></mrow></","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104174"},"PeriodicalIF":1.8,"publicationDate":"2024-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141593950","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 : 2024-06-27DOI: 10.1016/j.nonrwa.2024.104171
Claudio Pessoa , Ronisio Ribeiro
In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines, such that the linear differential systems that define the piecewise one have a center and two saddles. That is, the linear differential system in the region between the two parallel lines (called of central subsystem) has a center and the others subsystems have saddles. We prove that if the central subsystem has a real or a boundary center, then at least six limit cycles can bifurcate from the periodic annulus by linear perturbations. Four passing through the three zones and two passing through two zones. Now, if the central subsystem has a virtual center, then at leas four limit cycles can bifurcate from the periodic annulus by linear perturbations, three passing through the three zones and one passing through two zones. For this, we obtain a normal form for these piecewise Hamiltonian systems and study the number of zeros of its Melnikov functions defined in two and three zones.
{"title":"Bifurcation of limit cycles from a periodic annulus formed by a center and two saddles in piecewise linear differential system with three zones","authors":"Claudio Pessoa , Ronisio Ribeiro","doi":"10.1016/j.nonrwa.2024.104171","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104171","url":null,"abstract":"<div><p>In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by two parallel straight lines, such that the linear differential systems that define the piecewise one have a center and two saddles. That is, the linear differential system in the region between the two parallel lines (called of central subsystem) has a center and the others subsystems have saddles. We prove that if the central subsystem has a real or a boundary center, then at least six limit cycles can bifurcate from the periodic annulus by linear perturbations. Four passing through the three zones and two passing through two zones. Now, if the central subsystem has a virtual center, then at leas four limit cycles can bifurcate from the periodic annulus by linear perturbations, three passing through the three zones and one passing through two zones. For this, we obtain a normal form for these piecewise Hamiltonian systems and study the number of zeros of its Melnikov functions defined in two and three zones.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104171"},"PeriodicalIF":1.8,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481796","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 : 2024-06-26DOI: 10.1016/j.nonrwa.2024.104170
Lin Chang
In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg–de Vries–Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.
{"title":"Convergence rate toward shock wave under periodic perturbation for generalized Korteweg–de Vries–Burgers equation","authors":"Lin Chang","doi":"10.1016/j.nonrwa.2024.104170","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104170","url":null,"abstract":"<div><p>In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg–de Vries–Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104170"},"PeriodicalIF":1.8,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141482154","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 : 2024-06-26DOI: 10.1016/j.nonrwa.2024.104168
Minh Le
The objective is to investigate the global existence of solutions for degenerate chemotaxis systems with logistic sources in a two-dimensional domain. It is demonstrated that the inclusion of logistic sources can exclude the occurrence of blow-up solutions, even in the presence of superlinear growth in the cross-diffusion rate. Our proof relies on the application of elliptic and parabolic regularity in Orlicz spaces and variational approach.
{"title":"Global existence of solutions to some degenerate chemotaxis systems with superlinear growth in cross-diffusion rates and logistic sources","authors":"Minh Le","doi":"10.1016/j.nonrwa.2024.104168","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104168","url":null,"abstract":"<div><p>The objective is to investigate the global existence of solutions for degenerate chemotaxis systems with logistic sources in a two-dimensional domain. It is demonstrated that the inclusion of logistic sources can exclude the occurrence of blow-up solutions, even in the presence of superlinear growth in the cross-diffusion rate. Our proof relies on the application of elliptic and parabolic regularity in Orlicz spaces and variational approach.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104168"},"PeriodicalIF":1.8,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481795","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 : 2024-06-25DOI: 10.1016/j.nonrwa.2024.104166
Ting Luo, Weifeng Zhang
In this paper, the Cauchy problem for the coupled Ostrovsky equations with an initial value in the Sobolev spaces of lower order is considered. With the bilinear estimate, it is proved that the initial value problem is locally well-posed in for by using Bourgain spaces. Moreover, if , it is demonstrated that one of the nonlinear iteration from the initial data to the putative solutions is discontinuous with an argument on the high-to-low frequency. In this sense, it is then concluded that the coupled Ostrovsky equations is ill-posed in for .
{"title":"The local well-posedness of the coupled Ostrovsky system with low regularity","authors":"Ting Luo, Weifeng Zhang","doi":"10.1016/j.nonrwa.2024.104166","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104166","url":null,"abstract":"<div><p>In this paper, the Cauchy problem for the coupled Ostrovsky equations with an initial value in the Sobolev spaces <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>×</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> of lower order <span><math><mi>s</mi></math></span> is considered. With the bilinear estimate, it is proved that the initial value problem is locally well-posed in <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>×</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>></mo><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span> by using Bourgain spaces. Moreover, if <span><math><mrow><mi>s</mi><mo><</mo><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span>, it is demonstrated that one of the nonlinear iteration from the initial data to the putative solutions is discontinuous with an argument on the high-to-low frequency. In this sense, it is then concluded that the coupled Ostrovsky equations is ill-posed in <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>×</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo><</mo><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span>.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104166"},"PeriodicalIF":1.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481791","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 : 2024-06-25DOI: 10.1016/j.nonrwa.2024.104162
Stanislav Antontsev , Ivan Kuznetsov , Sergey Sazhenkov , Sergey Shmarev
We study the multi-dimensional Cauchy–Dirichlet problem for the -parabolic equation with a regular nonlinear minor term, which models a non-instantaneous but very rapid absorption with the -growth. The minor term depends on a positive integer parameter and, as , converges weakly to the expression incorporating the Dirac delta function, which, in turn, models an instant absorption at the initial moment. We prove that an infinitesimal initial layer, associated with the Dirac delta function, is formed as , and that the family of regular weak solutions of the original problem converges to the so-called ‘strong-weak’ solution of a two-scale microscopic–macroscopic model. Furthermore, the equation of the microstructure can be integrated explicitly, which leads in a number of cases to the purely macroscopic formulation for the -parabolic equation provided with the corrected initial data.
{"title":"Solutions of impulsive p(x,t)-parabolic equations with an infinitesimal initial layer","authors":"Stanislav Antontsev , Ivan Kuznetsov , Sergey Sazhenkov , Sergey Shmarev","doi":"10.1016/j.nonrwa.2024.104162","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104162","url":null,"abstract":"<div><p>We study the multi-dimensional Cauchy–Dirichlet problem for the <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>-parabolic equation with a regular nonlinear minor term, which models a non-instantaneous but very rapid absorption with the <span><math><mrow><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>-growth. The minor term depends on a positive integer parameter <span><math><mi>n</mi></math></span> and, as <span><math><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mi>∞</mi></mrow></math></span>, converges weakly<span><math><msup><mrow></mrow><mrow><mo>⋆</mo></mrow></msup></math></span> to the expression incorporating the Dirac delta function, which, in turn, models an instant absorption at the initial moment. We prove that an infinitesimal initial layer, associated with the Dirac delta function, is formed as <span><math><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mi>∞</mi></mrow></math></span>, and that the family of regular weak solutions of the original problem converges to the so-called ‘strong-weak’ solution of a two-scale microscopic–macroscopic model. Furthermore, the equation of the microstructure can be integrated explicitly, which leads in a number of cases to the purely macroscopic formulation for the <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span>-parabolic equation provided with the corrected initial data.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104162"},"PeriodicalIF":1.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481794","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 : 2024-06-24DOI: 10.1016/j.nonrwa.2024.104163
Yachun Li , Peng Lu , Liang Zhao
It was proved that Euler–Maxwell systems converge globally-in-time to Euler–Poisson systems near non-constant equilibrium states when the speed of light . In this paper, we establish the global-in-time error estimates between smooth solutions of Euler–Maxwell systems and those of Euler–Poisson systems near non-constant equilibrium states. The main difficulty lies in the singularity of the error variable for the electric field , so that more careful estimates for the time derivatives of error variables should be established. The proof takes good advantage of the anti-symmetric structure of the error system and an induction argument on the order of the derivatives.
有研究证明，当光速为c→∞时，Euler-Maxwell系统在非恒定平衡态附近与Euler-Poisson系统在时间上全局收敛。本文建立了欧拉-麦克斯韦系统光滑解与欧拉-泊松系统在非恒定平衡态附近的全局时间误差估计。主要难点在于电场 E 的误差变量的奇异性，因此需要对误差变量的时间导数进行更仔细的估计。证明很好地利用了误差系统的反对称结构和导数阶次的归纳论证。
{"title":"Global-in-time error estimates of non-relativistic limits for Euler–Maxwell system near non-constant equilibrium","authors":"Yachun Li , Peng Lu , Liang Zhao","doi":"10.1016/j.nonrwa.2024.104163","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104163","url":null,"abstract":"<div><p>It was proved that Euler–Maxwell systems converge globally-in-time to Euler–Poisson systems near non-constant equilibrium states when the speed of light <span><math><mrow><mi>c</mi><mo>→</mo><mi>∞</mi></mrow></math></span>. In this paper, we establish the global-in-time error estimates between smooth solutions of Euler–Maxwell systems and those of Euler–Poisson systems near non-constant equilibrium states. The main difficulty lies in the singularity of the error variable for the electric field <span><math><mi>E</mi></math></span>, so that more careful estimates for the time derivatives of error variables should be established. The proof takes good advantage of the anti-symmetric structure of the error system and an induction argument on the order of the derivatives.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104163"},"PeriodicalIF":1.8,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141482153","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 : 2024-06-24DOI: 10.1016/j.nonrwa.2024.104169
Siegfried Carl , Patrick Winkert
<div><p>We prove existence results for the parabolic double phase obstacle problem: Find <span><math><mrow><mi>u</mi><mo>∈</mo><mi>K</mi><mo>⊂</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> with <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>⋅</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> satisfying <span><span><span><math><mrow><mn>0</mn><mo>∈</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>A</mi><mi>u</mi><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>∂</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mtext>in</mtext><msubsup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>A</mi><mo>:</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>→</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∗</mo></mrow></msubsup></mrow></math></span> given by <span><span><span><math><mrow><mi>A</mi><mi>u</mi><mo>≔</mo><mo>−</mo><mo>div</mo><mfenced><mrow><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>+</mo><mi>μ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mrow></mfenced><mspace></mspace><mtext>for</mtext><mi>u</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo></mrow></math></span></span></span>is the double phase operator acting on <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>τ</mi><mo>;</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> denoting the associated Musielak–Orlicz Sobolev space with generalized homogeneous boundary values. The obstacle is represented by the closed convex set <span><math><mi>K</mi></math></span> with the obstacle function <span><math><mi>ψ</mi></math></span> through <span><span><span><math><mrow><mi>K</mi><mo>=</mo><mrow><mo>{</mo><mi>v</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mspace></mspace><mo>:</mo><mspace></mspace><mi>v</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mi>ψ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mtext>for a.a.</mtext><mspace></mspace><mrow><mo>(</mo><mi>x</mi><mo>,<
我们证明了抛物线双相障碍问题的存在性结果：在 u(⋅,0)=0 时找到 u∈K⊂X0 满足 0∈ut+Au+F(u)+∂IK(u)inX0∗ 的 u∈K⊂X0，其中，A：X0→X0∗，Au≔-div|∇u|p-2∇u+μ(x)|∇u|q-2∇uforu∈X0，是作用于 X0=Lp(0,τ;W01,H(Ω))的双相算子，W01,H(Ω)表示具有广义同质边界值的相关穆西拉克-奥利兹索博廖夫空间。障碍由封闭凸集 K 表示，障碍函数 ψ 通过 K={v∈X0:v(x,t)≤ψ(x,t)for a.a.(x,t)∈Q=Ω×(0,τ)} 表示，IK 是与 K 相关的指示函数，∂IK 表示其在凸分析意义上的次微分。
{"title":"Parabolic double phase obstacle problems","authors":"Siegfried Carl , Patrick Winkert","doi":"10.1016/j.nonrwa.2024.104169","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104169","url":null,"abstract":"<div><p>We prove existence results for the parabolic double phase obstacle problem: Find <span><math><mrow><mi>u</mi><mo>∈</mo><mi>K</mi><mo>⊂</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> with <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>⋅</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> satisfying <span><span><span><math><mrow><mn>0</mn><mo>∈</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>A</mi><mi>u</mi><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>∂</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>K</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mtext>in</mtext><msubsup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∗</mo></mrow></msubsup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>A</mi><mo>:</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>→</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∗</mo></mrow></msubsup></mrow></math></span> given by <span><span><span><math><mrow><mi>A</mi><mi>u</mi><mo>≔</mo><mo>−</mo><mo>div</mo><mfenced><mrow><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>+</mo><mi>μ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi></mrow></mfenced><mspace></mspace><mtext>for</mtext><mi>u</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo></mrow></math></span></span></span>is the double phase operator acting on <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>τ</mi><mo>;</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>H</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> denoting the associated Musielak–Orlicz Sobolev space with generalized homogeneous boundary values. The obstacle is represented by the closed convex set <span><math><mi>K</mi></math></span> with the obstacle function <span><math><mi>ψ</mi></math></span> through <span><span><span><math><mrow><mi>K</mi><mo>=</mo><mrow><mo>{</mo><mi>v</mi><mo>∈</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mspace></mspace><mo>:</mo><mspace></mspace><mi>v</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mi>ψ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mtext>for a.a.</mtext><mspace></mspace><mrow><mo>(</mo><mi>x</mi><mo>,<","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104169"},"PeriodicalIF":1.8,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1468121824001093/pdfft?md5=8c45c9bac4bc0752dc7ba149da99cd83&pid=1-s2.0-S1468121824001093-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141480008","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 : 2024-06-24DOI: 10.1016/j.nonrwa.2024.104153
Ting Zhang, Guanwei Chen
In the whole space , we study the existence of two standing waves for a class of matrix nonlinear Schrödinger equations with potentials by using variational methods, where the nonlinearities are sublinear or asymptotically linear at infinity. The novelties are as follows. (1) The matrix nonlinear equations are defined in the whole space . (2) The nonlinearities are composed of two mixed nonlinear terms with different growth conditions. (3) The weight function may be sign-changing. (4) Our results can be applied to many examples.
{"title":"Multiple standing waves of matrix nonlinear Schrödinger equations with mixed growth nonlinearities in RN","authors":"Ting Zhang, Guanwei Chen","doi":"10.1016/j.nonrwa.2024.104153","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104153","url":null,"abstract":"<div><p>In the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, we study the existence of two standing waves for a class of matrix nonlinear Schrödinger equations with potentials by using variational methods, where the nonlinearities are sublinear or asymptotically linear at infinity. The novelties are as follows. (1) The matrix nonlinear equations are defined in the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. (2) The nonlinearities are composed of two mixed nonlinear terms with different growth conditions. (3) The weight function may be sign-changing. (4) Our results can be applied to many examples.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104153"},"PeriodicalIF":1.8,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481793","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号