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Global solutions in tree dimensions for systems describing a chemotaxis phenomenon. (English) Zbl 1017.92005
From the introduction: We call chemotaxis the movement of a population under the influence of the chemical substances produced by this very population. This phenomenon appears, for example, when amoebas head through the slime towards the high concentration of a chemical substance they have secreted, called cyclic AMP. From the initial situation two phenomena can arise: either an excessive concentration of the population appears in some places or a homogeneous balance of cells develops. In this article we are interested in the mathematical study of the second effect for a parabolic-elliptic system describing this movement. The system we consider comes from the Keller and Segel model [see E. F. Keller and L. A. Segel, J. Theor. Biol. 30, 225-264 (1971)].
The following parabolic-elliptic system is generally introduced: \[ \begin{cases} \partial u/\partial t-d_1\Delta u=-\text{div} \bigl(u\nabla \psi(a) \bigr) \quad & \text{on }(0,+\infty) \times\Omega,\\ -\Delta a=\alpha (u-\overline u_0)\quad & \text{on }(0,+\infty) \times\Omega,\\ d_1(\partial u/ \partial n)-u\partial \psi(a)/ \partial n=0,\;\partial a/\partial n=0\quad & \text{on }(0,+\infty) \times\Gamma,\\ u(0,.)=u_0\quad & \text{on }\Omega, \end{cases} \tag{S} \] where the unknown \(u\) is always the cell density and the unknown \(a\) is translated as the concentration of chemotactic substance such as \(\overline a=0\). The function \(\psi\) always has the above properties and \(\alpha =k_1/d_2\) is a positive constant. Our aim is to determine some conditions leading to the existence and the uniqueness of a time-global solution to the system (S).

MSC:
92C17 Cell movement (chemotaxis, etc.)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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