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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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