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Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. (English) Zbl 1290.35139

Summary: We consider nonnegative solutions of the Neumann boundary value problem for the chemotaxis system \[ \begin{cases} u_t=\Delta u-\chi\nabla\cdot(u\nabla v), \quad &x\in\Omega,\;t>0,\\ \tau v_t=\Delta v-v+u, \quad &x\in\Omega,\;t>0. \end{cases} \]
in a smooth bounded convex domain \(\Omega\subset\mathbb R^n\), \(n\geq 1\), where \(\tau> 0\), \(\chi\in\mathbb R\) and \(f\) is a smooth function generalizing the logistic source \(f(s)=\kappa s-\mu s^2\), \(s\geq 0\), with \(\kappa >0\) and \(\mu >0\).
It is shown that if \(\mu\) is sufficiently large then for all sufficiently smooth initial data the problem possesses a unique global-in-time classical solution that is bounded in \(\Omega\times (0, \infty)\). Known results, asserting boundedness under the additional restriction \(n\leq 2\), are thereby extended to arbitrary space dimensions.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35A09 Classical solutions to PDEs
92C17 Cell movement (chemotaxis, etc.)
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