Pub Date : 2024-09-17DOI: 10.1016/j.nonrwa.2024.104220
Jun Wang, Xuan Wang
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":"Jun Wang, Xuan Wang","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":"81 ","pages":"Article 104220"},"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
Xiao Wang, Xin Xu
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":"Xiao Wang, Xin Xu","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":"81 ","pages":"Article 104207"},"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
Fan Wu , Guomei Zhao
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":"Fan Wu , Guomei Zhao","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":"81 ","pages":"Article 104217"},"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}
Pub Date : 2024-09-14DOI: 10.1016/j.nonrwa.2024.104216
Nadia Skoglund Taki
In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modeling the antiplane shear deformation of a static frictional adhesive contact problem. The adhesion process has been extensively studied, but it is usual to assume a priori that the intensity of adhesion is bounded by introducing truncation operators. The aim of this article is to remove this restriction.
The proof is based on an iterative approximation scheme showing that the problem has a unique solution. A key ingredient is finding uniform a priori bounds for each iterate. These are obtained by adapting versions of the Moser iteration to our system of equations.
{"title":"Weak solvability of elliptic variational inequalities coupled with a nonlinear differential equation","authors":"Nadia Skoglund Taki","doi":"10.1016/j.nonrwa.2024.104216","DOIUrl":"10.1016/j.nonrwa.2024.104216","url":null,"abstract":"<div><p>In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modeling the antiplane shear deformation of a static frictional adhesive contact problem. The adhesion process has been extensively studied, but it is usual to assume a priori that the intensity of adhesion is bounded by introducing truncation operators. The aim of this article is to remove this restriction.</p><p>The proof is based on an iterative approximation scheme showing that the problem has a unique solution. A key ingredient is finding uniform a priori bounds for each iterate. These are obtained by adapting versions of the Moser iteration to our system of equations.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104216"},"PeriodicalIF":1.8,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S146812182400155X/pdfft?md5=f0c8fe02474b074f403dd8ccda859969&pid=1-s2.0-S146812182400155X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142232681","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 : 2024-09-14DOI: 10.1016/j.nonrwa.2024.104214
Min Xiao , Jie Zhao , Qiurong He
<div><p>In this paper, we deal with the following Neumann-initial boundary value problem for a quasilinear chemotaxis model describing tumor angiogenesis: <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>D</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mi>χ</mi><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>u</mi><mo>∇</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>ξ</mi><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>u</mi><mo>∇</mo><mi>w</mi><mo>)</mo></mrow><mo>+</mo><mi>μ</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace><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><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>v</mi><mo>∇</mo><mi>w</mi><mo>)</mo></mrow><mo>−</mo><mi>v</mi><mo>+</mo><mi>u</mi><mo>,</mo><mspace></mspace><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><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>u</mi><mo>,</mo><mspace></mspace><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><mfrac><mrow><mi>∂</mi><mi>u</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>∂</mi><mi>v</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>∂</mi><mi>w</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>∂</mi><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mi>v</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>in a bounded smooth domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>n</mi><mo>≤</mo><mn>3</mn><mo>)</mo></mrow></mrow></math></span>, where the parameter <span><math><mrow><mi>χ</mi><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mi>ξ</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mi>μ</mi><mo>≥</mo><
{"title":"Large time behavior of solution to a quasilinear chemotaxis model describing tumor angiogenesis with/without logistic source","authors":"Min Xiao , Jie Zhao , Qiurong He","doi":"10.1016/j.nonrwa.2024.104214","DOIUrl":"10.1016/j.nonrwa.2024.104214","url":null,"abstract":"<div><p>In this paper, we deal with the following Neumann-initial boundary value problem for a quasilinear chemotaxis model describing tumor angiogenesis: <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>D</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mi>χ</mi><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>u</mi><mo>∇</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>ξ</mi><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>u</mi><mo>∇</mo><mi>w</mi><mo>)</mo></mrow><mo>+</mo><mi>μ</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace><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><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>v</mi><mo>∇</mo><mi>w</mi><mo>)</mo></mrow><mo>−</mo><mi>v</mi><mo>+</mo><mi>u</mi><mo>,</mo><mspace></mspace><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><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>u</mi><mo>,</mo><mspace></mspace><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><mfrac><mrow><mi>∂</mi><mi>u</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>∂</mi><mi>v</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>∂</mi><mi>w</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>∂</mi><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mi>v</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>in a bounded smooth domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>n</mi><mo>≤</mo><mn>3</mn><mo>)</mo></mrow></mrow></math></span>, where the parameter <span><math><mrow><mi>χ</mi><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mi>ξ</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mi>μ</mi><mo>≥</mo><","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104214"},"PeriodicalIF":1.8,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142230686","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.2024.104218
Xuan Tian , Shangjiang Guo
This paper is concerned with a size-structured diffusive predator–prey model with chemotaxis mechanism. The existence, linearized stability and monotonicity with respect to the growth rates of boundary steady-state solutions are analyzed. Moreover, the global stability of trivial steady-state solution under certain conditions is proved by constructing Lyapunov functional. We investigate the local and global bifurcations of positive steady-state solutions that emanate from semi-trivial steady-state solutions using Lyapunov–Schmidt reduction and bifurcation techniques when the fertility intensity of a predator or prey is used as a bifurcation parameter. It is shown that the nonlinear nonlocal chemotaxis term can lead to the emergence of Allee effect.
{"title":"Dynamics of a size-structured predator–prey model with chemotaxis mechanism","authors":"Xuan Tian , Shangjiang Guo","doi":"10.1016/j.nonrwa.2024.104218","DOIUrl":"10.1016/j.nonrwa.2024.104218","url":null,"abstract":"<div><p>This paper is concerned with a size-structured diffusive predator–prey model with chemotaxis mechanism. The existence, linearized stability and monotonicity with respect to the growth rates of boundary steady-state solutions are analyzed. Moreover, the global stability of trivial steady-state solution under certain conditions is proved by constructing Lyapunov functional. We investigate the local and global bifurcations of positive steady-state solutions that emanate from semi-trivial steady-state solutions using Lyapunov–Schmidt reduction and bifurcation techniques when the fertility intensity of a predator or prey is used as a bifurcation parameter. It is shown that the nonlinear nonlocal chemotaxis term can lead to the emergence of Allee effect.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104218"},"PeriodicalIF":1.8,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142232680","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 : 2024-09-11DOI: 10.1016/j.nonrwa.2024.104215
M. Marras , S. Vernier-Piro , T. Yokota
<div><p>We study a new class of Keller–Segel models, which presents a limited flux and an optimal transport of cells density according to chemical signal density. As a prototype of this class we study radially symmetric solutions to the parabolic–elliptic system <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mspace></mspace><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mfrac><mrow><mi>u</mi><mo>∇</mo><mi>u</mi></mrow><mrow><msqrt><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow><mo>−</mo><mi>χ</mi><msub><mrow><mi>k</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mfrac><mrow><mi>u</mi><mo>∇</mo><mi>v</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>v</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mspace></mspace><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><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>μ</mi><mo>+</mo><mi>u</mi><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><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 no flux boundary conditions in a ball <span><math><mrow><mi>B</mi><mo>=</mo><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> and initial condition <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn><mo>,</mo><mi>χ</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><msub><mrow><mi>k</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>μ</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mi>d</mi><mi>x</mi><mo>.</mo></mrow></math></span> Under suitable conditions on <span><math><mi>α</mi></math></span> and <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> it is shown that the solution blows up in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-norm at a finite time <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></sp
{"title":"Behavior in time of solutions to a degenerate chemotaxis system with flux limitation","authors":"M. Marras , S. Vernier-Piro , T. Yokota","doi":"10.1016/j.nonrwa.2024.104215","DOIUrl":"10.1016/j.nonrwa.2024.104215","url":null,"abstract":"<div><p>We study a new class of Keller–Segel models, which presents a limited flux and an optimal transport of cells density according to chemical signal density. As a prototype of this class we study radially symmetric solutions to the parabolic–elliptic system <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mspace></mspace><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mfrac><mrow><mi>u</mi><mo>∇</mo><mi>u</mi></mrow><mrow><msqrt><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow><mo>−</mo><mi>χ</mi><msub><mrow><mi>k</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mfrac><mrow><mi>u</mi><mo>∇</mo><mi>v</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>v</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mspace></mspace><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><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>μ</mi><mo>+</mo><mi>u</mi><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><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 no flux boundary conditions in a ball <span><math><mrow><mi>B</mi><mo>=</mo><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> and initial condition <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn><mo>,</mo><mi>χ</mi><mo>></mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><msub><mrow><mi>k</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> and <span><math><mrow><mi>μ</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mi>d</mi><mi>x</mi><mo>.</mo></mrow></math></span> Under suitable conditions on <span><math><mi>α</mi></math></span> and <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> it is shown that the solution blows up in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-norm at a finite time <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></sp","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104215"},"PeriodicalIF":1.8,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1468121824001548/pdfft?md5=6f13fb993e83f3c2deb3af5f4ca4b00b&pid=1-s2.0-S1468121824001548-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142169088","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 : 2024-09-11DOI: 10.1016/j.nonrwa.2024.104208
Qianying Zhang , Mingxin Wang
In this paper, we study an SIR epidemic model with nonlocal diffusion and double free boundaries, which can be used to describe a class of biological phenomena: the depletion of native resources by all individuals, the infected individuals do not lose their fertility completely, the recovered individuals are immune and no longer infected, the infected and recovered individuals spread along the same free boundary. We first investigate the existence and uniqueness of global solution, long time behaviors and some sufficient conditions for spreading and vanishing. Then we estimate the spreading speed and derive that accelerated spreading could happen when the kernel function does not satisfy a threshold condition.
本文研究了一个具有非局部扩散和双重自由边界的 SIR 流行病模型，该模型可用于描述一类生物现象：所有个体的本地资源耗尽，受感染个体不会完全丧失生育能力，康复个体具有免疫力且不再受感染，受感染个体和康复个体沿同一自由边界扩散。我们首先研究了全局解的存在性和唯一性、长时间行为以及扩散和消失的一些充分条件。然后，我们估算了传播速度，并推导出当核函数不满足阈值条件时，可能会发生加速传播。
{"title":"Dynamics for a nonlocal diffusive SIR epidemic model with double free boundaries","authors":"Qianying Zhang , Mingxin Wang","doi":"10.1016/j.nonrwa.2024.104208","DOIUrl":"10.1016/j.nonrwa.2024.104208","url":null,"abstract":"<div><p>In this paper, we study an SIR epidemic model with nonlocal diffusion and double free boundaries, which can be used to describe a class of biological phenomena: the depletion of native resources by all individuals, the infected individuals do not lose their fertility completely, the recovered individuals are immune and no longer infected, the infected and recovered individuals spread along the same free boundary. We first investigate the existence and uniqueness of global solution, long time behaviors and some sufficient conditions for spreading and vanishing. Then we estimate the spreading speed and derive that accelerated spreading could happen when the kernel function does not satisfy a threshold condition.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104208"},"PeriodicalIF":1.8,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142169148","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 : 2024-09-11DOI: 10.1016/j.nonrwa.2024.104213
Nikolaos S. Papageorgiou , Zijia Peng
We consider a Dirichlet problem driven by the double phase differential operator and a parametric reaction which has the combined effects of a singular term and of a convective perturbation. Using nonlinear operators of monotone type, truncation and comparison techniques, and fixed point theory, we show that for all small values of the parameter, the problem has a bounded positive solution.
{"title":"Singular double phase problems with convection","authors":"Nikolaos S. Papageorgiou , Zijia Peng","doi":"10.1016/j.nonrwa.2024.104213","DOIUrl":"10.1016/j.nonrwa.2024.104213","url":null,"abstract":"<div><p>We consider a Dirichlet problem driven by the double phase differential operator and a parametric reaction which has the combined effects of a singular term and of a convective perturbation. Using nonlinear operators of monotone type, truncation and comparison techniques, and fixed point theory, we show that for all small values of the parameter, the problem has a bounded positive solution.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104213"},"PeriodicalIF":1.8,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142169149","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 : 2024-09-10DOI: 10.1016/j.nonrwa.2024.104210
Zijia Peng , Yining Zhao , Fengzhen Long
This paper is concerned with an evolutionary variational hemivariational inequality which is considered in the form of a nonlinear evolution inclusion. In the inclusion, both the convex subdifferential and Clarke subdifferential are related to the time derivative of the unknown function. In addition, the convex subdifferential operator is unbounded and thus the Signorini case is included. Due to these features, the existing surjectivity theorems for evolution inclusions are not applicable. Instead, the Rothe method based on the temporal discretization strategy is used to study the solvability of this new variational hemivariational inequality. We first show the existence of solutions to the discrete stationary problem. Then we establish a convergence result of the semidiscrete scheme and prove the existence and uniqueness of the solution to the inclusion. Moreover, we show the existence and uniqueness of the solution to the original variational hemivariational inequality. Finally, an example is given to illustrate the abstract result.
{"title":"Existence and uniqueness of the solution to a new class of evolutionary variational hemivariational inequalities","authors":"Zijia Peng , Yining Zhao , Fengzhen Long","doi":"10.1016/j.nonrwa.2024.104210","DOIUrl":"10.1016/j.nonrwa.2024.104210","url":null,"abstract":"<div><p>This paper is concerned with an evolutionary variational hemivariational inequality which is considered in the form of a nonlinear evolution inclusion. In the inclusion, both the convex subdifferential and Clarke subdifferential are related to the time derivative of the unknown function. In addition, the convex subdifferential operator is unbounded and thus the Signorini case is included. Due to these features, the existing surjectivity theorems for evolution inclusions are not applicable. Instead, the Rothe method based on the temporal discretization strategy is used to study the solvability of this new variational hemivariational inequality. We first show the existence of solutions to the discrete stationary problem. Then we establish a convergence result of the semidiscrete scheme and prove the existence and uniqueness of the solution to the inclusion. Moreover, we show the existence and uniqueness of the solution to the original variational hemivariational inequality. Finally, an example is given to illustrate the abstract result.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"81 ","pages":"Article 104210"},"PeriodicalIF":1.8,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142164238","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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