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Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. (English) Zbl 1190.92004
Summary: We consider the classical parabolic-parabolic E. F. Keller and L. A. Segel system [J. Theor. Biol. 26, No. 3, 399–415 (1970; Zbl 1170.92306)]
\[ u_t= \Delta u-\nabla\cdot(u\nabla v), \qquad v_t=\Delta v-v+u, \quad x\in\Omega,\;t>0, \]
under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega\subset\mathbb R^n\). It is proved that in space dimensions \(n\geq 3\), for each \(q>n/2\) and \(p>n\) one can find \(\varepsilon_0>0\) such that if the initial data \((u_0,v_0)\) satisfy \(\|u_0\|_{L^q(\Omega)}>\varepsilon\) and \(\|\nabla v_0\|_{L^p(\Omega)}<\varepsilon\) then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations. In particular, \((u,v)\) approaches the steady state \((m,m)\) as \(t\to\infty\), where \(m\) is the total mass \(m:=\int_\Omega u_0\) of the population.
Moreover, we show that if \(\Omega\) is a ball then for arbitrary prescribed \(m>0\) there exist unbounded solutions emanating from initial data \((u_0,v_0)\) having total mass \(\int_\Omega u_0=m\).

MSC:
92C17 Cell movement (chemotaxis, etc.)
35B40 Asymptotic behavior of solutions to PDEs
35K35 Initial-boundary value problems for higher-order parabolic equations
35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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