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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)
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