In this paper, we consider a doubly degenerate taxis system of the form \(\begin{aligned} \left\{ \begin{array}{rllc} u_{t}& =\nabla \cdot \big (uv\nabla u\big )-\nabla \cdot \big (u^2v\nabla v\big ),\ & x\in \Omega ,& t>0,\\ v_{t}& =\Delta v-f(u)v,\ & x\in \Omega ,& t>0, \end{array}\right. \end{aligned}\) in a smoothly bounded domain \(\Omega \subset \mathbb {R}^N\) , \(N\ge 2\) . The function \(f\in C^1([0,\infty ))\) is assumed to satisfy the constraints \(\begin{aligned} f(s)\ge c_f s^\alpha \quad \text {for all }s\ge 1\quad \text {and}\quad f(s)\le C_f s^\alpha \quad \text {for all }s>0 \end{aligned}\) for \(\alpha \in (0,\tfrac{2}{N})\) and constants \(C_f,c_f>0\) . We establish the existence of a global continuous weak solution which remains bounded for all times, without any condition on the initial data beyond regularity assumptions. Moreover, we obtain convergence toward \((u_\infty ,0)\) in the large-time limit, where the limit function \(u_\infty =w(\cdot ,1)\) is provided by the weak solution of the associated parabolic PDE \(\begin{aligned} w_{\tau }=\nabla \cdot \big (a(x,\tau )w\nabla w\big )-\nabla \cdot \big (b(x,\tau )w^2\big ),\quad x\in \Omega ,\ \tau \in (0,1), \end{aligned}\) where the coefficients \(a(x,\tau )\) and \(b(x,\tau )\) are defined intrinsically as appropriately rescaled quantities of v and \(v\nabla v\) , respectively.
