boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source
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2016
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Abstract
This article concerns the attraction-repulsion chemotaxis system with
nonlinear diffusion and logistic source,
$$\displaylines{
u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla\cdot(\chi u\nabla v)
+\nabla\cdot(\xi u\nabla w)+ru-\mu u^\eta, \cr
x\in\Omega,\; t>0,\cr
v_t=\Delta v+\alpha u-\beta v, \quad x\in\Omega, \; t>0,\cr
w_t=\Delta w+\gamma u-\delta w, \quad x\in\Omega,\; t>0
}$$
under Neumann boundary conditions in a bounded domain
$\Omega\subset\mathbb{R}^3$ with smooth boundary.
We show that if the diffusion is strong enough or
the logistic dampening is sufficiently powerful, then the corresponding
system possesses a global bounded classical solution for any sufficiently
regular initial data. Moreover, it is proved that if $r=0$,
$\beta>\frac{1}{2(\eta-1)}$ and $\delta>\frac{1}{2(\eta-1)}$ for the latter case,
then $u(\cdot,t)\to 0$, $ v(\cdot,t)\to 0$ and $ w(\cdot,t)\to 0$ in
$L^\infty(\Omega)$ as $ t \to \infty$.
| Reference Key |
wang2016electronicboundedness
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|---|---|
| Authors | ;Yilong Wang |
| Journal | icsoft 2006 - 1st international conference on software and data technologies, proceedings |
| Year | 2016 |
| DOI |
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