The doubly degenerate nutrient taxis model $\left\{\begin{array}{cc}{u}_t=\nabla \cdot (uv\nabla u)-\nabla \cdot (u^{2}v\nabla v)+\ell uv,\hfill & x\in \Omega ,t>0,\hfill \\ v_t=\Delta v-uv,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$is considered in smoothly bounded convex subdomains of the plane, with $\ell \ge 0$. It is shown that for any $p>2$ and each fixed nonnegative $u_0\in W^{1,\infty }\left(\Omega \right)$, a smallness condition exclusively involving $v_0$ can be identified as sufficient to ensure that an associated no-flux type initial–boundary value problem with $(u,v){|}_{t=0}=(u_0,v_0)$ admits a global weak solution satisfying $\mathrm{ess}{sup}_{t>0}{\Vert u(\cdot ,t)\Vert }_{L^p\left(\Omega \right)}<\infty$. The proof relies on the use of an apparently novel class of functional inequalities which provide estimates from below for certain Dirichlet integrals involving possibly degenerate weight functions.

双简并营养出租车模型 $\left\{\begin{array}{cc}{你}_{吨}=\nabla \cdot (你v\nabla 你)-\nabla \cdot ({你}^{2}v\nabla v)+\ell 你v,\hfill & X\in \Omega ,吨>0,\hfill \\ v_{吨}=\Delta v-你v,\hfill & X\in \Omega ,吨>0,\hfill \end{array}\right.$在平面的平滑有界凸子域中考虑，其中 $\ell \ge 0$. 表明对于任意$磷>2$ 和每个固定的非负 ${你}_0\in {宽}^{1,\infty }\left(\Omega \right)$，一个完全涉及的小条件 $v_0$ 可以被识别为足以确保相关的无通量类型初始边界值问题与 $(你,v){|}_{吨=0}=({你}_0,v_0)$ 承认一个全局弱解满足 $\mathrm{ess}{超}_{吨>0}{\Vert 你(\cdot ,吨)\Vert }_{{升}^{磷}\left(\Omega \right)}<\infty$. 该证明依赖于使用明显新颖的一类函数不等式，这些不等式为某些涉及可能退化权重函数的狄利克雷积分提供了从下面的估计。