Global boundedness in a fully parabolic quasilinear chemotaxis system with singular sensitivity (Q1706538)

In this paper, the author studies the global boundedness of solutions to the quasilinear fully parabolic chemotaxis system: \[ \begin{cases} u_t =\nabla\cdot(D(u)\nabla u -S(u)\nabla \phi(v)), \\ v_t =\Delta v-v+u, \end{cases} \] in a bounded domain \(\Omega\subset\mathbb{R}^n\) (\(n\geq 2\)) subject to the non-flux boundary conditions, the diffusivity fulfills \(D(u) = a_0(u +1)^{-\alpha}\) with \(a_0 > 0\) and \(\alpha\geq 0\), while the density-signal governed sensitivity satisfies \(0\leq S(u)\leq b_0(u +1)^\beta\) and \(0 < \phi'(v) \leq\frac{\chi}{v^k}\) for \(b_0, \chi > 0\) and \(\beta, k\in \mathbb{R}\). It is shown that the solution is globally bounded provided \(\alpha+\beta < 1\) and \(k \leq 1\). This result demonstrates the effect of signal-dependent sensitivity on the blow-up prevention.