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Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity. (English) Zbl 1329.35011
The authors study classical solutions to the parabolic-elliptic Keller-Segel system \begin{alignedat}{2} u_t & = \Delta u - \nabla \cdot (u \chi (v) \nabla v), \qquad & (x,t) &\in \Omega \times (0,\infty), \\ 0&= \Delta v - v+u, \qquad & (x,t) &\in \Omega \times (0,\infty),\end{alignedat} endowed with homogeneous Neumann boundary conditions and initial data $$u_0$$, where $$\Omega \subset \mathbb{R}^n$$ is a bounded domain with smooth boundary, $$n \geq 2$$, and $$u_0 \in C^0 (\bar{\Omega})$$ is nonnegative with $$u_0 \not\equiv 0$$. In addition, the chemotactic sensitivity function $$\chi \in C^1((0,\infty))$$ is assumed to satisfy $0 < \chi (s) \leq {\chi_0 \over s^k} \qquad\text{for all } s \in [\gamma, \infty)$ with constants $$k \geq 1$$, $$\chi_0 >0$$, and $$\gamma = \gamma ( \|u_0\|_{L^1 (\Omega)}, \text{diam} (\Omega)) >0$$. The authors establish the existence of a unique global and bounded classical solution provided that either $$k=1$$ and $$\chi_0 < {2 \over n}$$ or $$k >1$$ and $$\chi_0 < {2 \over n} \cdot {k^k \over (k-1)^{k-1}} \gamma^{k-1}$$ is fulfilled. An important ingredient of the proof is the derivation of an a priori positive lower bound for $$v$$. The global existence as well as the uniform boundedness of $$u$$ in $$L^\infty (\Omega)$$ is finally inferred from a bound on $$u$$ in $$L^p (\Omega)$$ for some $$p > {n \over 2}$$. Thereby, the authors extend results from P. Biler [Adv. Math. Sci. Appl. 9, No. 1, 347–359 (1999; Zbl 0941.35009)] and T. Nagai and T. Senba [Adv. Math. Sci. Appl. 8, No. 1, 145–156 (1998; Zbl 0902.35010)], where global existence of weak or radial solutions was studied for $$\chi(v) = \frac{\chi_0}{v}$$, but the uniform boundedness of solutions remained an open problem.

##### MSC:
 35A09 Classical solutions to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35K55 Nonlinear parabolic equations 92C17 Cell movement (chemotaxis, etc.) 35K59 Quasilinear parabolic equations
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