On an attraction-repulsion chemotaxis system with a logistic source. (English) Zbl 1336.35338
Summary: This paper is devoted to the attraction-repulsion chemotaxis system with a logistic source:
\[
\begin{cases}
u_t= \Delta u- \chi \nabla (u \nabla v)+ \mu \nabla (u \nabla w) + \mathcal{R}(u), \; x\in \Omega,\;t>0,\\
v_t= \Delta v- \alpha_1 v + \beta_1 u, \; x\in \Omega,\;t>0\\
w_t=\Delta w-\alpha_2 w + \beta_2 u, \; x\in \Omega,\;t>0 \end{cases}
\]
where \(\Omega \subset \mathbb {R}^N(N\geqslant 1)\) is a bounded domain with smooth boundary and \(\mathcal {R}(s)\leqslant a-bs^\tau \). For the case \(\varrho =0\), we show that when the repulsion prevails over the attraction in the sense that \(\mu \beta _2-\chi \beta _1>0\), there exist global bounded classical solutions for any logistic damping \(\tau \geqslant 1\). When the attraction dominates the repulsion in the sense that \(\mu \beta _2-\chi \beta _1<0\), the classical solutions are still global and bounded provided that the logistic damping is strong. For the case \(\varrho >0\), we will investigate the similar problem for \(N=1\) and \(N=2\). We will also study the regularity of stationary solutions.
where \(\Omega \subset \mathbb {R}^N(N\geqslant 1)\) is a bounded domain with smooth boundary and \(\mathcal {R}(s)\leqslant a-bs^\tau \). For the case \(\varrho =0\), we show that when the repulsion prevails over the attraction in the sense that \(\mu \beta _2-\chi \beta _1>0\), there exist global bounded classical solutions for any logistic damping \(\tau \geqslant 1\). When the attraction dominates the repulsion in the sense that \(\mu \beta _2-\chi \beta _1<0\), the classical solutions are still global and bounded provided that the logistic damping is strong. For the case \(\varrho >0\), we will investigate the similar problem for \(N=1\) and \(N=2\). We will also study the regularity of stationary solutions.
MSC:
35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
92C17 | Cell movement (chemotaxis, etc.) |
35B41 | Attractors |
35B65 | Smoothness and regularity of solutions to PDEs |
35A01 | Existence problems for PDEs: global existence, local existence, non-existence |