Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
IF 3.2
1区 数学
Q1 MATHEMATICS
Guoqiang Ren, Xing Zhou
求助PDF
{"title":"Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity","authors":"Guoqiang Ren, Xing Zhou","doi":"10.1515/anona-2023-0125","DOIUrl":null,"url":null,"abstract":"\n <jats:p>In this study, we investigate the two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity: <jats:disp-formula id=\"j_anona-2023-0125_eq_001\">\n <jats:alternatives>\n <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0125_eq_001.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\n <m:mfenced open=\"{\" close=\"\">\n <m:mrow>\n <m:mtable displaystyle=\"true\">\n <m:mtr>\n <m:mtd columnalign=\"left\">\n <m:msub>\n <m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>t</m:mi>\n </m:mrow>\n </m:msub>\n <m:mo>=</m:mo>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mo>⋅</m:mo>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:msup>\n <m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>θ</m:mi>\n <m:mo>−</m:mo>\n <m:mn>1</m:mn>\n </m:mrow>\n </m:msup>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mo>−</m:mo>\n <m:mi>χ</m:mi>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mo>⋅</m:mo>\n <m:mfenced open=\"(\" close=\")\">\n <m:mrow>\n <m:mfrac>\n <m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>v</m:mi>\n </m:mrow>\n </m:mfrac>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mi>v</m:mi>\n </m:mrow>\n </m:mfenced>\n <m:mo>,</m:mo>\n </m:mtd>\n <m:mtd columnalign=\"left\">\n <m:mi>x</m:mi>\n <m:mo>∈</m:mo>\n <m:mi mathvariant=\"normal\">Ω</m:mi>\n <m:mo>,</m:mo>\n <m:mspace width=\"0.33em\" />\n <m:mi>t</m:mi>\n <m:mo>></m:mo>\n <m:mn>0</m:mn>\n <m:mo>,</m:mo>\n </m:mtd>\n </m:mtr>\n <m:mtr>\n <m:mtd columnalign=\"left\">\n <m:msub>\n <m:mrow>\n <m:mi>v</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>t</m:mi>\n </m:mrow>\n </m:msub>\n <m:mo>=</m:mo>\n <m:mi mathvariant=\"normal\">Δ</m:mi>\n <m:mi>v</m:mi>\n <m:mo>−</m:mo>\n <m:mi>v</m:mi>\n <m:mo>+</m:mo>\n <m:mi>u</m:mi>\n <m:mo>+</m:mo>\n <m:mi>g</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>x</m:mi>\n <m:mo>,</m:mo>\n <m:mi>t</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mo>,</m:mo>\n </m:mtd>\n <m:mtd columnalign=\"left\">\n <m:mi>x</m:mi>\n <m:mo>∈</m:mo>\n <m:mi mathvariant=\"normal\">Ω</m:mi>\n <m:mo>,</m:mo>\n <m:mspace width=\"0.33em\" />\n <m:mi>t</m:mi>\n <m:mo>></m:mo>\n <m:mn>0</m:mn>\n <m:mo>,</m:mo>\n </m:mtd>\n </m:mtr>\n <m:mtr>\n <m:mtd columnalign=\"left\" />\n </m:mtr>\n </m:mtable>\n </m:mrow>\n </m:mfenced>\n <m:mspace width=\"2.0em\" />\n <m:mspace width=\"2.0em\" />\n <m:mspace width=\"2.0em\" />\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mo>∗</m:mo>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n </m:math>\n <jats:tex-math>\\left\\{\\begin{array}{ll}{u}_{t}=\\nabla \\cdot \\left({u}^{\\theta -1}\\nabla u)-\\chi \\nabla \\cdot \\left(\\frac{u}{v}\\nabla v\\right),& x\\in \\Omega ,\\hspace{0.33em}t\\gt 0,\\\\ {v}_{t}=\\Delta v-v+u+g\\left(x,t),& x\\in \\Omega ,\\hspace{0.33em}t\\gt 0,\\\\ \\end{array}\\right.\\hspace{2.0em}\\hspace{2.0em}\\hspace{2.0em}\\left(\\ast )</jats:tex-math>\n </jats:alternatives>\n </jats:disp-formula> in a bounded domain with smooth boundary. We present the global boundedness of weak solutions to the model (<jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0125_eq_002.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mo>∗</m:mo>\n </m:math>\n <jats:tex-math>\\ast </jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>) if <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0125_eq_003.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>θ</m:mi>\n <m:mo>></m:mo>\n <m:mfrac>\n <m:mrow>\n <m:mn>3</m:mn>\n </m:mrow>\n <m:mrow>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:mfrac>\n </m:math>\n <jats:tex-math>\\theta \\gt \\frac{3}{2}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> and (1.10)–(1.11). This result improves our recent work.</jats:p>","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/anona-2023-0125","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
引用
批量引用
Abstract
In this study, we investigate the two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity:
u
t
=
∇
⋅
(
u
θ
−
1
∇
u
)
−
χ
∇
⋅
u
v
∇
v
,
x
∈
Ω
,
t
>
0
,
v
t
=
Δ
v
−
v
+
u
+
g
(
x
,
t
)
,
x
∈
Ω
,
t
>
0
,
(
∗
)
\left\{\begin{array}{ll}{u}_{t}=\nabla \cdot \left({u}^{\theta -1}\nabla u)-\chi \nabla \cdot \left(\frac{u}{v}\nabla v\right),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta v-v+u+g\left(x,t),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ \end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\ast )
in a bounded domain with smooth boundary. We present the global boundedness of weak solutions to the model (
∗
\ast
) if
θ
>
3
2
\theta \gt \frac{3}{2}
and (1.10)–(1.11). This result improves our recent work.
具有非线性扩散和奇异敏感性的二维趋化系统中的全局有界性
在本研究中,我们研究了具有非线性扩散和奇异敏感性的二维趋化系统: u t = ∇ ⋅ ( u θ - 1∇ u ) - χ ∇ ⋅ u v∇ v , x ∈ Ω , t > 0 , v t = Δ v - v + u + g ( x , t ) , x ∈
本文章由计算机程序翻译,如有差异,请以英文原文为准。
来源期刊
期刊介绍:
Advances in Nonlinear Analysis (ANONA) aims to publish selected research contributions devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The Journal focuses on papers that address significant problems in pure and applied nonlinear analysis. ANONA seeks to present the most significant advances in this field to a wide readership, including researchers and graduate students in mathematics, physics, and engineering.