GLOBAL BOUNDEDNESS IN A QUASILINEAR CHEMOTAXIS-CONSUMPTION SYSTEM WITH SIGNAL-DEPENDENT MOTILITY AND SUPER-QUADRATIC DAMPING

In this paper, we consider a quasilinear chemotaxis-consumption model <p class="disp_formula">$ \left\{ \begin{array}{lll} u_t=\Delta (v^{\alpha}u^{m})+ru-\mu u^{l},\quad &amp;x\in\Omega,\; t&gt;0,\\ v_t=\Delta v-uv, &amp;x\in\Omega,\; t&gt;0 \end{array} \right. $ within a smoothly bounded domain $ \Omega\subset\mathbb{R}^n $ under homogeneous Neumann boundary conditions, where the parameters $ \alpha, r, \mu&gt;0 $ and $ l, m&gt;1 $. For any sufficiently regular initial data and parameters $ l, m&gt;1 $ with $ l&gt;m+1 $, it is shown that the aforementioned system possesses at least one global weak solution with a boundedness property <p class="disp_formula">$ \|u(\cdot,t)\|_{L^{p}(\Omega)}+\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C $ for all $ p\geq 2 $ and $ t&gt;0 $. This finding indicates the regularizing effect of super-quadratic damping of a logistic-type source under strong degeneracy of signal-dependent motility, even though the cross-diffusion is simultaneously enhanced.