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 $$\displaylines{ u_t =\Delta u-\nabla\cdot(u|\nabla v|^{p-2}\nabla v),\quad x\in\Omega,\; t>0,\cr v_t =\Delta v-v+u,\quad x\in\Omega,\; t>0,\cr \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial\nu...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2020-12-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2020/122/abstr.html |
| Summary: | We consider the Keller-Segel system with gradient dependent chemotactic sensitivity
$$\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 $1<p<n/(n-1)$. |
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| ISSN: | 1072-6691 |