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$$ {\mathbb{R}}^N...
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| Published in: | Mathematical methods in the applied sciences Vol. 46; no. 8; pp. 9252 - 9267 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Freiburg
Wiley Subscription Services, Inc
30.05.2023
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| Subjects: | |
| ISSN: | 0170-4214, 1099-1476 |
| Online Access: | Get full text |
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| Summary: | In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “
u$$ 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 $$. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0170-4214 1099-1476 |
| DOI: | 10.1002/mma.9050 |