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Global behaviour of a reaction-diffusion system modelling chemotaxis. (English) Zbl 0918.35064
By a change of unknown functions in a partial differential system of reaction-diffusion type, which models chemotaxis, it is obtained the system \[ u_t= \Delta u-\nabla\cdot(u\nabla v),\quad v_t= \alpha\Delta v-\beta v+ \gamma(u-1)\quad\text{on }\mathbb{R}_+\times \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\geq 2\), with piecewise smooth boundary \(\Gamma= \partial\Omega\); \(\alpha\), \(\beta\), \(\gamma\) are positive constants. This system is completed by the initial and boundary conditions \[ u(0,.)= u_0,\quad v(0,.)= v_0\quad\text{on }\Omega;\quad \nu\nabla u= \nu\nabla u= 0, \] on \(\mathbb{R}_+\times\Gamma\), where \(\nu\) is the outer normal on \(\Gamma\), so it models the dynamics of a population (concentration \(u\)) moving in \(\Omega\), driven by gradient of chemotactic agents (concentration \(v\)) produced by the population.
The existence and uniqueness of a local weak solution is proved. The main aim of this paper is the study of the global behavior of solutions, using two Lyapunov functionals.

MSC:
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
92C40 Biochemistry, molecular biology
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
92C15 Developmental biology, pattern formation
92D25 Population dynamics (general)
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