On the existence of solutions for a parabolic‐elliptic chemotaxis model with flux limitation and logistic source
In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “ u$$ u $$” and a chemical stimulus “ v$$ v $$” in a bounded and regular domain Ω$$ \Omega $$ of ℝN$$ {\mathbb{R}}^N...
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| Veröffentlicht in: | Mathematical methods in the applied sciences Jg. 46; H. 8; S. 9252 - 9267 |
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| Sprache: | Englisch |
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30.05.2023
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| ISSN: | 0170-4214, 1099-1476 |
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| Abstract | In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “
u$$ u $$” and a chemical stimulus “
v$$ v $$” in a bounded and regular domain
Ω$$ \Omega $$ of
ℝN$$ {\mathbb{R}}^N $$. The equation for
u$$ u $$ is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as
−χdiv(u|∇v|p−2∇v),$$ -\chi \operatorname{div}\left(u{\left|\nabla v\right|}^{p-2}\nabla v\right), $$
for
p>1$$ p>1 $$. The chemical substance distribution
v$$ v $$ satisfies the elliptic equation
−Δv+v=u.$$ -\Delta v+v=u. $$
The evolution of
u$$ u $$ is also determined by a logistic type growth term
μu(1−u)$$ \mu u\left(1-u\right) $$. The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for
p<3/2$$ p<3/2 $$ and any
N≥2$$ N\ge 2 $$. |
|---|---|
| AbstractList | In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “
” and a chemical stimulus “
” in a bounded and regular domain
of
. The equation for
is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as
for
. The chemical substance distribution
satisfies the elliptic equation
The evolution of
is also determined by a logistic type growth term
. The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for
and any
. In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “ u$$ u $$” and a chemical stimulus “ v$$ v $$” in a bounded and regular domain Ω$$ \Omega $$ of ℝN$$ {\mathbb{R}}&#x0005E;N $$. The equation for u$$ u $$ is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as −χdiv(u|∇v|p−2∇v),$$ -\chi \operatorname{div}\left(u{\left&#x0007C;\nabla v\right&#x0007C;}&#x0005E;{p-2}\nabla v\right), $$ for p>1$$ p>1 $$. The chemical substance distribution v$$ v $$ satisfies the elliptic equation −Δv+v=u.$$ -\Delta v&#x0002B;v&#x0003D;u. $$ The evolution of u$$ u $$ is also determined by a logistic type growth term μu(1−u)$$ \mu u\left(1-u\right) $$. The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for p<3/2$$ p<3/2 $$ and any N≥2$$ N\ge 2 $$. In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “u$$ u $$” and a chemical stimulus “v$$ v $$” in a bounded and regular domain Ω$$ \Omega $$ of ℝN$$ {\mathbb{R}}^N $$. The equation for u$$ u $$ is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as −χdiv(u|∇v|p−2∇v),$$ -\chi \operatorname{div}\left(u{\left|\nabla v\right|}^{p-2}\nabla v\right), $$for p>1$$ p>1 $$. The chemical substance distribution v$$ v $$ satisfies the elliptic equation −Δv+v=u.$$ -\Delta v+v=u. $$The evolution of u$$ u $$ is also determined by a logistic type growth term μu(1−u)$$ \mu u\left(1-u\right) $$. The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for p<3/2$$ p<3/2 $$ and any N≥2$$ N\ge 2 $$. |
| Author | Tello, Jose Ignacio Sastre‐Gomez, Silvia |
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| Cites_doi | 10.1080/03605302.2021.1975132 10.1007/s00332‐010‐9082‐x 10.1016/j.jtbi.2015.07.023 10.5802/aif.204 10.1016/0022‐5193(70)90092‐5 10.1016/0022‐5193(71)90050‐6 10.1016/j.jde.2016.05.008 10.1016/j.aml.2020.106351 10.1007/978-3-642-61798-0 10.1016/j.jde.2018.01.040 10.1016/j.jde.2016.07.008 10.1512/iumj.2022.71.9042 10.1371/journal.pcbi.1000890 10.1016/j.jmaa.2022.126376 10.1007/s00285-008-0201-3 10.1080/03605302.2016.1277237 10.1090/btran/17 10.1016/0009‐2509(89)85098‐5 10.1016/j.jde.2015.07.019 10.1016/j.jde.2019.05.026 10.1080/03605300701319003 10.1080/03605302.2020.1712417 10.4171/rmi/1132 10.1016/j.aml.2022.108299 10.3934/krm.2012.5.51 10.1007/s10440-019-00275-z |
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| Title | On the existence of solutions for a parabolic‐elliptic chemotaxis model with flux limitation and logistic source |
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