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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
##### Keywords:
chemotaxis; global existence; boundedness; blow-up
Full Text:
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