Refined existence theorems for doubly degenerate chemotaxis–consumption systems with large initial data

This work considers the doubly degenerate nutrient model \(\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot \left( u^{m-1}v\nabla u\right) -\nabla \cdot \left( f(u)v\nabla v\right) +\ell uv, & x\in \Omega ,\,t>0,\\&v_t=\Delta v-uv, & x\in \Omega ,\,t>0, \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& u_t=\mathrm{\nabla }·\left(u^{m-1},v,\mathrm{\nabla },u\right)-\mathrm{\nabla }·\left(f,(,u,),v,\mathrm{\nabla },v\right)+\ell uv,\hfill & \hfill x\in \mathrm{\Omega },t>0,\\ \hfill & v_t=\mathrm{\Delta }v-uv,\hfill & \hfill x\in \mathrm{\Omega },t>0,\end{array}\right)\end{array}$$ under no-flux boundary conditions in a smoothly bounded convex domain \(\Omega \subset \mathbb R^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ( \(n\le 2\) $n\le 2$ ), where the nonnegative function \(f\in C^1([0,\infty ))\) $f\in C^1([0,\infty ))$ is assumed to satisfy \(f(s)\le C_fs^{\alpha }\) $f(s)\le C_f{s}^{\alpha }$ with \(\alpha >0\) $\alpha >0$ and \(C_f>0\) $C_f>0$ for all \(s\ge 1\) $s\ge 1$ . When \(m=2\) $m=2$ , it was shown that a global weak solution exists, either in one-dimensional setting with \(\alpha =2\) $\alpha =2$ , or in two-dimensional version with \(\alpha \in \left( 1,\frac{3}{2}\right) \) $\alpha \in \left(1,,,\frac{3}{2}\right)$ . The main results in this paper assert the global existence of weak solutions for \(1\le m<3\) $1\le m<3$ and classical solutions for \(3\le m<4\) $3\le m<4$ to the above system under the assumption \(\begin{aligned} \alpha \in \left\{ \begin{aligned}&\left[ m-1,\min \left\{ m,\frac{m}{2}+1\right\} \right] ~~ & \text {if}~~n=1,\quad \quad \text {and}\\&\left( m-1,\min \left\{ m,\frac{m}{2}+1\right\} \right) ~~ & \text {if}~~n=2, \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \alpha \in \left\{\begin{array}{ccc}& \left[m,-,1,,,min,\left\{m,,,\frac{m}{2},+,1\right\}\right]\hfill & \hfill \text{if}n=1,\text{and}\\ \hfill & \left(m,-,1,,,min,\left\{m,,,\frac{m}{2},+,1\right\}\right)\hfill & \hfill \text{if}n=2,\end{array}\right)\end{array}$$ which extend the range \(\alpha \in (1,\frac{3}{2})\) $\alpha \in (1,\frac{3}{2})$ to \(\alpha \in (1,2)\) $\alpha \in (1,2)$ in two dimensions for the case \(m=2\) $m=2$ . Our proof will be based on a new observation on the coupled energy-type functional and on an inequality with general form.