A dimension-independent critical exponent in a nutrient taxis system

In a ball $\Omega\subset R^n$ with arbitrary $n\ge 1$, the chemotaxis-consumption system $$\left\{ \begin{array}{l} u_t = \nabla \cdot \big(D(u)\nabla u\big) - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - uv, \end{array} \right.$$ is considered under no-flux boundary conditions for $u$, and for prescribed constant positive boundary data for $v$. Under the assumption that $D\in C^3([0,\infty))$ satisfies $$D(\xi)\ge k_D (\xi+1)^{-\alpha} \qquad \mbox{for all } \xi\ge 0 \qquad \qquad (\star)$$ with some $\alpha<1$ and some $k_D>0$, it is shown that for each nonnegative and radially symmetric $u_0\in \bigcup_{q>\max\{2,n\}} W^{1,q}(\Omega)$, a uniquely determined global bounded classical solution exists. This complements a previous result according to which given any positive $D\in C^3([0,\infty))$ fulfilling $D(\xi) \le K_D (\xi+1)^{-\alpha}$ with some $\alpha>1$ and $K_D>0$, one can find nonnegative radial initial data $u_0\in C_0^\infty(\Omega)$ such that no global solution exists.