Global weak solutions to a doubly degenerate nutrient taxis system on the whole real line

Abstract

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The paper studies a one-dimensional doubly degenerate nutrient-taxis system on the whole real line that models pattern formation in bacterial populations. It formulates the Cauchy problem for coupled PDEs governing bacterial density 𝑢 u and nutrient concentration 𝑣 v, and proves global existence of weak solutions under suitable regularity and integrability conditions on the initial data. To handle both the degenerate cross-diffusion and the unbounded domain, the author introduces regularized problems on bounded intervals ( − 1 𝜀 , 1 𝜀 ) (− ε 1 ​

, ε 1 ​

) and derives uniform (in 𝜀 ε) estimates. Compactness arguments—culminating in an application of the Aubin–Lions lemma—are then used to pass to the limit and construct global weak solutions, stated as the main result (Theorem 1.1). The work situates the model within nutrient-taxis and Keller–Segel–type literature and outlines the analytical challenges specific to the doubly degenerate structure and whole-space setting.