The degenerate migration-consumption model \(\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta (\phi (v)u)+au-bu^\gamma ,\quad & x\in \Omega ,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega ,\quad t>0, \end{array} \right. \end{aligned}\) is considered under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^n\) ( \(n\ge 1\) ), where the parameters \(a>0,b>0\) , \(\gamma >1\) , and the motility function \(\phi \) suitably generalizes the prototype given by \(\phi (s)=s^\alpha \) for all \(s\ge 0\) with some \(\alpha >0\) . Consequently, the main obstacle of analysis results from a possible degeneracy at small signal concentrations. When \(\phi \) is suitably smooth with \(\alpha \ge \frac{1}{\frac{n}{2}+\gamma -1}\) , in two- and higher-dimensional settings, it is shown that if one of the following cases holds: (i) \(\gamma >2\) ; (ii) \(\gamma =2\) and b is sufficiently large, then for all appropriately regular initial data this system possesses global classical solutions. In particular, for any \(b>0\) , the same conclusion remains valid, irrespective of whether \(\alpha \ge 1\) and \(\gamma =2\) in two-dimensional counterpart or \(\alpha >0\) and \(\gamma >1\) in one-dimensional case. In the case when \(\gamma =2\) , for arbitrary \(b>0\) , by means of distinct approaches, it is asserted that in both cases \(\alpha \in (0,\frac{1}{2})\) and \(\alpha \ge \frac{1}{2}\) this system admits globally defined weak solutions for all \(n\ge 2\) . Furthermore, in the case when \(\gamma =2\) , for all \(\alpha >1\) and \(n\ge 3\) , then the above weak solutions eventually become smooth and classical. To the best of our knowledge, it seems to be the first rigorous analytical results concerned with global classical solvability for arbitrary dimensions and ultimate smoothness of global weak solutions in degenerate cases for all \(n\ge 3.\)
