Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity
We consider the Keller-Segel system with gradient dependent chemotactic sensitivity $$\displaylines{ u_t =\Delta u-\nabla\cdot(u|\nabla v|^{p-2}\nabla v),\quad x\in\Omega,\; t>0,\cr v_t =\Delta v-v+u,\quad x\in\Omega,\; t>0,\cr \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial\nu...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2020-12-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2020/122/abstr.html |
| _version_ | 1818617080101994496 |
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| author | Jianlu Yan Yuxiang Li |
| author_facet | Jianlu Yan Yuxiang Li |
| author_sort | Jianlu Yan |
| collection | DOAJ |
| description | We consider the Keller-Segel system with gradient dependent chemotactic sensitivity
$$\displaylines{
u_t =\Delta u-\nabla\cdot(u|\nabla v|^{p-2}\nabla v),\quad x\in\Omega,\; t>0,\cr
v_t =\Delta v-v+u,\quad x\in\Omega,\; t>0,\cr
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial\nu}=0,\quad
x\in\partial\Omega,\; t>0,\cr
u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \quad x\in\Omega
}$$
in a smooth bounded domain $\Omega\subset \mathbb{R}^n$, $n\geq2$.
We shown that for all reasonably regular initial data $u_0\geq 0$ and $v_0\geq0$,
the corresponding Neumann initial-boundary value problem possesses a global weak
solution which is uniformly bounded provided that $1<p<n/(n-1)$. |
| first_indexed | 2024-12-16T17:00:00Z |
| format | Article |
| id | doaj.art-1c6832f4dba04c2d8375aa905fdad188 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-16T17:00:00Z |
| publishDate | 2020-12-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-1c6832f4dba04c2d8375aa905fdad1882022-12-21T22:23:46ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912020-12-012020122,114Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivityJianlu Yan0Yuxiang Li1 Southeast Univ., Nanjing, China Southeast Univ., Nanjing, China We consider the Keller-Segel system with gradient dependent chemotactic sensitivity $$\displaylines{ u_t =\Delta u-\nabla\cdot(u|\nabla v|^{p-2}\nabla v),\quad x\in\Omega,\; t>0,\cr v_t =\Delta v-v+u,\quad x\in\Omega,\; t>0,\cr \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial\nu}=0,\quad x\in\partial\Omega,\; t>0,\cr u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \quad x\in\Omega }$$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$, $n\geq2$. We shown that for all reasonably regular initial data $u_0\geq 0$ and $v_0\geq0$, the corresponding Neumann initial-boundary value problem possesses a global weak solution which is uniformly bounded provided that $1<p<n/(n-1)$.http://ejde.math.txstate.edu/Volumes/2020/122/abstr.htmlkeller-segel systemweak solution chemotactic sensitivity |
| spellingShingle | Jianlu Yan Yuxiang Li Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity Electronic Journal of Differential Equations keller-segel system weak solution chemotactic sensitivity |
| title | Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity |
| title_full | Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity |
| title_fullStr | Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity |
| title_full_unstemmed | Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity |
| title_short | Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity |
| title_sort | existence and boundedness of solutions for a keller segel system with gradient dependent chemotactic sensitivity |
| topic | keller-segel system weak solution chemotactic sensitivity |
| url | http://ejde.math.txstate.edu/Volumes/2020/122/abstr.html |
| work_keys_str_mv | AT jianluyan existenceandboundednessofsolutionsforakellersegelsystemwithgradientdependentchemotacticsensitivity AT yuxiangli existenceandboundednessofsolutionsforakellersegelsystemwithgradientdependentchemotacticsensitivity |