A strongly degenerate migration-consumption model in domains of arbitrary dimension

TL;DR

This paper investigates a degenerate migration-consumption PDE model in convex domains of arbitrary dimension, establishing the existence of global weak solutions with bounded entropy for certain parameter ranges.

Contribution

It introduces a generalized model with a function  that extends a prototype power law, and proves the existence of global weak solutions in all dimensions for specific  ranges.

Findings

Existence of global weak solutions for  in (0,1) and [1,2].

Solutions have bounded entropy over time.

Entropy remains uniformly bounded for  in [1,2].

Abstract

In a smoothly bounded convex domain $\Omega\subset R^n$ with $n\ge 1$, a no-flux initial-boundary value problem for \[ \left\{ \begin{array}{l} u_t=\Delta \big(u\phi(v)\big), v_t=\Delta v-uv, \end{array} \right. \] is considered under the assumption that near the origin, the function $\phi$ suitably generalizes the prototype given by \[ \phi(\xi)=\xi^\alpha, \qquad \xi\in [0,\xi_0]. \] By means of separate approaches, it is shown that in both cases $\alpha\in (0,1)$ and $\alpha\in [1,2]$ some global weak solutions exist which, inter alia, satisfy $C(T):= {\rm esssup} {}_{t\in (0,T)} \int_\Omega u(\cdot,t)\ln u(\cdot,t) < \infty$ for all $T>0$, with $\sup_{T>0} C(T)<\infty$ if $\alpha\in [1,2]$.