Weak global solvability of a doubly degenerate parabolic-elliptic nutrient taxis system

TL;DR

This paper proves the existence of global weak solutions for a complex doubly degenerate nutrient taxis system, overcoming analytical challenges with a regularization approach and advanced compactness techniques.

Contribution

It introduces a novel regularization method and a Harnack-type inequality to establish global weak solutions for a challenging degenerate PDE system.

Findings

Existence of global weak solutions for the system.

Development of a Harnack-type inequality for the second equation.

Application of compactness theorems to handle degeneracy.

Abstract

This work studies the following doubly degenerate parabolic-elliptic nutrient taxis system $$ \begin{cases} u_t = (uvu_x)_x -(u^2 vv_x)_x + uv, \\[1.5 ex] \hspace{0.2 cm}0 = v_{xx} - uv + f(x,t), \end{cases} $$ in a bounded interval $\Omega \subset \mathbb{R}$, under no-flux boundary conditions and nonnegative initial value $u(x,0) = u_0(x) \geq 0$, where $f(x,t) \geq 0$ is known external supply of the nutrient. It is shown that for any nonnegative $u_0 \in W^{1,\infty}(\Omega)$ and $f \in C^1\big(\bar{\Omega} \times [0,\infty) \big)$, $f \not \equiv 0$, a global weak solution of the problem can be constructed by means of a regularization approach. The core of the analysis lies on a Harnack-type inequality for the second that allows us to overcome the lack of uniform coercivity. Together with time regularity properties, we obtain relative compactness through a combination of the Arzel\`a-Ascoli theorem and the Aubin-Lions lemma.