Proliferation-dominated large time behaviour in a planar doubly degenerate nutrient taxis system

This study is concerned with nonnegative solutions of the no-flux initial-boundary value problem for the doubly degenerate nutrient taxis system with a logistic source, as given by \begin{align*}\left\{\begin{array}{ll}u_t = \nabla\cdot (uv\nabla u) - \nabla\cdot (u^2 v\nabla v)+ u - u^2,\[1mm]v_t = \Delta v -uv,\end{array} \right.\end{align*} in a smoothly bounded planar domain $\Omega$ with $(u,v)|_{t=0}=(u_0,v_0)$ . It is shown that despite substantially weakened diffusion, the stabilizing effects of logistic-type cell kinetics may overbalance any heterogeneity-supporting tendency of cross-diffusion: Namely, it is seen that for all suitably smooth initial data satisfying $v_0 \gt 0$ in $\overline{\Omega}$ and \begin{align*}\int_\Omega \ln u_0~ \gt ~-\infty,\end{align*} a corresponding continuous weak solution, known to exist globally according to previous literature, has the property that as $t\to\infty$ , \begin{align*} u(\cdot,t) \rightharpoonup 1 \quad \text{in } L^1(\Omega) \qquad\text{and}\qquad v(\cdot,t)\to 0 \quad \text{in } L^\infty(\Omega). \end{align*}