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

This work considers the doubly degenerate nutrient model {equation*}{AH1} \{ {split} &u_t=∇·(uᵐ⁻¹v∇ u)-∇·(f(u)v∇ v)+ uv,&&x∈Ω,\,t>0, &v_t=Δ v-uv, &&x∈Ω,\,t>0, {split} . {equation*} under no-flux boundary conditions in a smoothly bounded convex domain Ω⊂ Rⁿ (n≤ 2), where the nonnegative function f∈ C¹([0,∞)) is assumed to satisfy f(s)≤ Cfsα with α>0 and Cf>0 for all s≥ 1. When $m=2$, it was shown that a global weak solution exists, either in one-dimensional setting with α=2, or in two-dimensional version with α∈(1,3/2). The main results in this paper assert the global existence of weak solutions for 1≤ m<3 and classical solutions for 3≤ m<4 to the above system under the assumption {equation*} α∈ \{ {split} &[m-1,min\{m,m/2+1\}]~~&&if~~n=1, &(m-1,min\{m,m/2+1\})~~&&if~~n=2, {split} . {equation*} which extend the range α∈(1,3/2) to α∈(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.