$ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity

This paper studies the parabolic-parabolic Keller-Segel system with supercritical sensitivity: $u_{t}=\nabla\cdot(\phi (u) \nabla u)-\nabla \cdot(\varphi(u)\nabla v)$, $v_{t}=\Delta v -v+u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^n$ $(n\ge2)$, the diffusivity fulfills $\phi(u)\ge a_0(u+1)^{\gamma}$ with $\gamma\ge0$ and $a_0>0$, while the chemotactic sensitivity satisfies $0\le \varphi(u)\le b_0u(u+1)^{\alpha+\gamma-1}$ with $\alpha>\frac{2}{n}$ and $b_0>0$. It is proved that the problem possesses a globally bounded solution for $\frac{4}{n+2}<\alpha<2$, whenever $\|u_0\|_{L^{\frac{n\alpha}{2}}(\Omega)}$ and $\|\nabla v_0\|_{L^{\frac{n\alpha+2\gamma}{2-\alpha}}(\Omega)}$ is sufficiently small. Similarly, the above conclusion still holds for $\alpha>2$ provided that $\|u_{0}\|_{L^{n\alpha-n}(\Omega)}$ and $\|\nabla v_0\|_{L^{\infty}(\Omega)}$ are small enough.