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

This work considers the doubly degenerate nutrient model u t = ∇ · u m - 1 v ∇ u - ∇ · f (u) v ∇ v + ℓ u v , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under no-flux boundary conditions in a smoothly bounded convex domain Ω ⊂ R n ( n ≤ 2 ), where the nonnegative function f ∈ C 1 ([ 0 , ∞)) is assumed to satisfy f (s) ≤ C f s α with α > 0 and C f > 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 α ∈ m - 1 , min m , m 2 + 1 if n = 1 , and m - 1 , min m , m 2 + 1 if n = 2 , 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. [ABSTRACT FROM AUTHOR]