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The doubly degenerate nutrient taxis system(0.1)

\begin{equation} \left \{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u^{\alpha }v\nabla v)+\ell uv,&x\in \Omega ,\, t\gt 0,\\[5pt] & v_{t}=\Delta v-uv,&x\in \Omega ,\, t\gt 0,\\ \end{aligned} \right . \end{equation}

is considered under zero-flux boundary conditions in a smoothly bounded domain

$\Omega \subset \mathbb{R}^3$ where

$\alpha \gt 0,\chi \gt 0$ and

$\ell \gt 0$. By developing a novel class of functional inequalities to address the challenges posed by the doubly degenerate diffusion mechanism in (0.1), it is shown that for

$\alpha \in (\frac {3}{2},\frac {19}{12})$, the associated initial-boundary value problem admits a global continuous weak solution for sufficiently regular initial data. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium