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

Abstract

This work considers the doubly degenerate nutrient model \begin{equation*}\label{AH1} \left\{ \begin{split} &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{split} \right. \end{equation*} under no-flux boundary conditions in a smoothly bounded convex domain $\Omega\subset \mathbb{R}^n$ ($n\le 2$), where the nonnegative function $f\in C^1([0,\infty))$ is assumed to satisfy $f(s)\le C_fs^{\alpha}$ with $\alpha>0$ and $C_f>0$ for all $s\ge 1$. When $m=2$, it was shown that a global weak solution exists, either in one-dimensional setting with $\alpha=2$, or in two-dimensional version with $\alpha\in(1,\frac{3}{2})$. The main results in this paper assert the global existence of weak solutions for $1\le m<3$ and classical solutions for $3\le m<4$ to the above system under the assumption \begin{equation*} \alpha\in \left\{ \begin{split} &\left[m-1,\min\left\{m,\frac{m}{2}+1\right\}\right]~~&&\textrm{if}~~n=1,\quad\quad\textrm{and} &\left(m-1,\min\left\{m,\frac{m}{2}+1\right\}\right)~~&&\textrm{if}~~n=2, \end{split} \right. \end{equation*} which extend the range $\alpha\in(1,\frac{3}{2})$ to $\alpha\in(1,2)$ in two dimensions for the case $m=2$. Our proof will be based on a new observation on the coupled energy-type functional and on an inequality with general form.