Pub Date : 2024-09-14DOI: 10.1016/j.nonrwa.2024.104216
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":"","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":null,"pages":null},"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
<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":"","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":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":"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
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":"","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":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":"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
<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":"","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":null,"pages":null},"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
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":"","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":null,"pages":null},"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
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":"","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":null,"pages":null},"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
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":"","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":null,"pages":null},"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}
Pub Date : 2024-09-09DOI: 10.1016/j.nonrwa.2024.104206
Ocean gyres are modelled by the two-dimensional vorticity equation for inviscid flow on a rotating sphere, since their flow is governed by the tangential velocity components, whereas the vertical velocity component is negligible. From the vorticity equation we derive nonlinear elliptic equations for the square of the meridional velocity component as well as for the azimuthal velocity component. Using maximum principles we then show that, under suitable conditions on the oceanic vorticity, the velocity extrema are attained on the boundary of a subtropical gyre located in the zonal band between 15 and 45 Northern, respectively Southern latitude, where the five major gyres are found.
{"title":"Velocity extrema in ocean gyre flows","authors":"","doi":"10.1016/j.nonrwa.2024.104206","DOIUrl":"10.1016/j.nonrwa.2024.104206","url":null,"abstract":"<div><p>Ocean gyres are modelled by the two-dimensional vorticity equation for inviscid flow on a rotating sphere, since their flow is governed by the tangential velocity components, whereas the vertical velocity component is negligible. From the vorticity equation we derive nonlinear elliptic equations for the square of the meridional velocity component as well as for the azimuthal velocity component. Using maximum principles we then show that, under suitable conditions on the oceanic vorticity, the velocity extrema are attained on the boundary of a subtropical gyre located in the zonal band between 15<span><math><msup><mrow></mrow><mrow><mo>∘</mo></mrow></msup></math></span> and 45<span><math><msup><mrow></mrow><mrow><mo>∘</mo></mrow></msup></math></span> Northern, respectively Southern latitude, where the five major gyres are found.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142164237","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 : 2024-09-07DOI: 10.1016/j.nonrwa.2024.104209
This paper is concerned with the Dirichlet initial boundary value problem of an epitaxial thin film growth equation involving gradient-type logarithmic nonlinearity and absorption terms. By introducing an equivalent norm and approximating Lipschitz functions, combining with the technique of Faedo–Galerkin approximation and the family of potential wells, we establish the well-posedness of global weak solutions. Meantime, we classify the decay properties and grow-up phenomenon of the considered problem by using the energy functional and the related Nehari manifold.
{"title":"Asymptotic behaviors of global weak solutions for an epitaxial thin film growth equation","authors":"","doi":"10.1016/j.nonrwa.2024.104209","DOIUrl":"10.1016/j.nonrwa.2024.104209","url":null,"abstract":"<div><p>This paper is concerned with the Dirichlet initial boundary value problem of an epitaxial thin film growth equation involving gradient-type logarithmic nonlinearity and absorption terms. By introducing an equivalent norm and approximating Lipschitz functions, combining with the technique of Faedo–Galerkin approximation and the family of potential wells, we establish the well-posedness of global weak solutions. Meantime, we classify the decay properties and grow-up phenomenon of the considered problem by using the energy functional and the related Nehari manifold.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142150111","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 : 2024-09-07DOI: 10.1016/j.nonrwa.2024.104212
This paper studies the upper bound of the lifespan of classical solutions of the initial value problems for one dimensional wave equations with quasilinear terms of space-, or time-derivatives of the unknown function. The result for the space-derivative case guarantees the optimality of the general theory for nonlinear wave equations, and its proof is carried out by combination of ordinary differential inequality and iteration method on the lower bound of the weighted functional of the solution.
{"title":"Blow-up of classical solutions of quasilinear wave equations in one space dimension","authors":"","doi":"10.1016/j.nonrwa.2024.104212","DOIUrl":"10.1016/j.nonrwa.2024.104212","url":null,"abstract":"<div><p>This paper studies the upper bound of the lifespan of classical solutions of the initial value problems for one dimensional wave equations with quasilinear terms of space-, or time-derivatives of the unknown function. The result for the space-derivative case guarantees the optimality of the general theory for nonlinear wave equations, and its proof is carried out by combination of ordinary differential inequality and iteration method on the lower bound of the weighted functional of the solution.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1468121824001512/pdfft?md5=8d3c17560b8046622b27bc7b02e5e3fe&pid=1-s2.0-S1468121824001512-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142150214","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}
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