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

Abstract

This work addresses the one-dimensional Cauchy problem for the doubly degenerate nutrient taxis model \begin{equation*} \begin{cases} \displaystyle \frac{\partial u}{\partial t} = \frac{\partial}{\partial x}(u v u_x) - \frac{\partial}{\partial x}(u^2 v v_x) + u v, & x\in \mathbb{R}, ~t>0, \\ \displaystyle \frac{\partial v}{\partial t} = \frac{\partial^2 v}{\partial x^2} - u v, & x\in \mathbb{R}, ~t>0, \\ u(x,0) = u_0(x) \geq 0, \quad v(x,0) = v_0(x)>0, ~ & x\in \mathbb{R}, \end{cases} \end{equation*} which models pattern formation in bacterial populations. The global existence of weak solutions is established for initial data satisfying appropriate regularity and integrability conditions. To account for the degeneracy caused by $u_0$ not being strictly positive and the difficulties arising from the unboundedness of the domain, we consider a family of regularized problems posed on bounded intervals $(-\frac{1}{\varepsilon}, \frac{1}{\varepsilon})$, for $\varepsilon \in (0,1)$. Through adequate estimates uniform in $\varepsilon$, we construct global solutions to the system by passing to the limit using the Aubin-Lions lemma.