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Finite dimensional attractor for one-dimensional Keller-Segel equations. (English) Zbl 1145.37337

Existence of a finite-dimensional attractor is established for the one-dimensional Keller-Segel chemotaxis model. Recall that the Keller-Segel equations (KS) describe the aggregation of cellular slime mold by a chemotactic process and read \[ \begin{aligned} \partial_t u &= a\partial_x^2 u - \partial_x(u\partial_x(\chi(\rho)) ), \qquad &&(x,t)\in I\times (0,+\infty),\\ \partial_t \rho &= b\partial_x^2 \rho + cu - d\rho, &&(x,t)\in I\times (0,+\infty),\end{aligned} \] where \(I=(\alpha,\beta)\) is a bounded interval and \(a,b,c,d\) are positive real numbers. This system is supplemented with homogeneous Neumann boundary conditions and initial data \((u_0,\rho_0)\in K_ l\), where \[ K_ l = \left\{ u_0\in L^2(I), \,u_0\geq 0,\, \| u_0\|_{L^1(I)} = l \text{ and } \rho_0\in H^1(I),\, \rho_0>0 \right\} \] and \( l\) is an arbitrary positive real number. Under general assumptions on the sensitivity function \(\chi\), the well-posedness of (KS) is shown together with several bounds which are uniform with respect to \(t\in (0,+\infty)\) and initial data in \(K_ l\). A bounded absorbing set is then constructed in \(K_ l\) and it is checked that the reaction terms in (KS) enjoy the properties needed to use abstract results and deduce the existence of an exponential attractor. The last section of the paper considers the case \(\chi(\rho)=k\rho\) for some \(k>0\), for which a Lyapunov functional is available. Additional information is then obtained on the \(\omega\)-limit sets of the solutions by classical arguments.

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41 Attractors
35K57 Reaction-diffusion equations
92C17 Cell movement (chemotaxis, etc.)
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