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Global attractor for a chemotaxis model with prevention of overcrowding. (English) Zbl 1065.35072
The author studies the following system of two parabolic equations modelling the chemotaxis with prevention of overcrowding: $\begin{cases} \frac{\partial u}{\partial t}=\nabla.(d_u(q(u)-q'(u)u)\nabla u-uq(u)\chi(v)\nabla v)+uf(u),\\ \frac{\partial v}{\partial t}=d_v\Delta v-g_1(u)-vg_2(v),\end{cases} \tag{1}$ which is considered in a bounded domain of $$\mathbb R^n$$ equipped by Robin boundary conditions. Here $$d_u$$ and $$d_v$$ are positive diffusion coefficients, and $$q(u)$$, $$g_1(u)$$, $$f(u)$$, $$g_2(v)$$ and $$\chi(v)$$ are assumed to satisfy the assumptions which gives the nondegenerate parabolicity of the problem considered and guarantee that the domain $$V_M:=\{v\geq0,\;\;0\leq u\leq U_M\}$$ (for some fixed positive $$U_M$$) is invariant with respect to the flow generated by (1). To be more precise, the last assumption requires: $$g_1(u)\geq0,\;\;f(U_M)\leq 0,\;\;q(U_M)=0$$ and, for the nondegeneracy, $$d(\xi):=d_u(q(\xi)-q'(\xi)\xi)>0,\;\;\xi\in[0,U_M].$$ Then, under some more technical assumptions on the nonlinearities, the author verifies the existence of a global attractor $$\mathcal A$$ for equation (1) in the domain $$V_M$$ (equipped with the appropriate topology).

##### MSC:
 35B41 Attractors 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 35K50 Systems of parabolic equations, boundary value problems (MSC2000)
##### Keywords:
volume filling model; quasilinear parabolic system
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##### References:
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