| Abstract: |
| In this talk, we study the doubly degenerate nutrient taxis system
\begin{align}
\begin{cases}\tag{$\star$}\label{eq-0.1}
u_t=\nabla \cdot(u v \nabla u)-\chi \nabla \cdot\left(u^{2} v \nabla v\right)+ u v, \ v_t=\Delta v-u v
\end{cases}
\end{align}
in a smooth bounded domain $\Omega \subset \mathbb{R}^2$, where $\chi>0$. We prove that for all reasonably regular initial data, the corresponding homogeneous Neumann initial-boundary value problem for \eqref{eq-0.1} possesses a global bounded weak solution which is continuous in its first and essentially smooth in its second component. There exists $u_{\infty}\in L^{\infty}(\Omega)$ such that this solution possesses the convergence property that
\begin{align}\label{eq-0.2}
u(t) \stackrel{*}{\rightharpoonup} u_{\infty}\quad \text{ and } \quad v(t) \rightarrow 0 \quad \text { in } L^{\infty}(\Omega) \quad \text { as } t \rightarrow \infty. \tag{$ \star \star$}
\end{align}
Furthermore, the limit $u_{\infty}$ in \eqref{eq-0.2} exhibits spatially heterogeneous under a criterion on the initial smallness of the second component. |
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