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A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production

Y. Tao acknowledges support of the National Natural Science Foundation of China (No. 11571070). M. Winkler was supported by the Deutsche Forschungsgemeinschaft within the project Analysis of chemotactic cross-diffusion in complex frameworks.
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  • We consider the chemotaxis-haptotaxis system

    in a bounded convex domain $ \Omega\subset \mathbb{R} ^n $ with smooth boundary, where $ \chi, \xi, \mu $ and $ \eta $ are positive constants, and where $ f \in C^1([0,\infty)) $ is a given function fulfilling $ f(0) \ge 0 $ and

    $ \begin{eqnarray*} f(s) \le K_f (s+1)^\alpha \qquad \mbox{for all } s\ge 0 \end{eqnarray*} $

    with some $ K_f >0 $ and $ \alpha>0 $.

    It is asserted that whenever

    $ \begin{eqnarray*} \alpha < \left\{ \begin{array}{ll} \frac{3}{2} \qquad & \mbox{if } n = 1, \\ \frac{n+6}{2(n+2)} \qquad & \mbox{if } n\ge 2, \end{array} \right. \end{eqnarray*} $

    the Neumann boundary problem with suitably regular initial data possesses a unique global and bounded classical solution.

    Mathematics Subject Classification: Primary: 35A01, 35B65, 35K57, 35Q92; Secondary: 92C17.

    Citation:

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