Global generalized solutions to a forager-exploiter model with superlinear degradation and their eventual regularity properties

Abstract

In this article we consider a cascaded taxis model for two proliferating and degrading species which thrive on the same nutrient but orient their movement according to different schemes. In particular, we assume the first group, the foragers, to orient their movement directly along an increasing gradient of the food density, while the second group, the exploiters, instead track higher densities of the forager group. Specifically, we will investigate an initial boundary-value problem for a prototypical forager-exploiter model of the form \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}l@{\quad }l@{\quad}l@{\quad}l@{\,}c} u_{t}&=&\Delta u-\nabla\cdot\big(u\nabla w\big)+f(u),\ &x\in\Omega,& t>0,\\ v_{t}&=&\Delta v-\nabla\cdot\big(v\nabla u\big)+g(v),\ &x\in\Omega,& t>0,\\ w_{t}&=&\Delta w-(u+v)w-\mu w+r(x,t),\ &x\in\Omega,& t>0, \end{array}\right. \end{align*}%} in a smoothly bounded domain $\Omega\subset\mathbb{R}^2$, where $\mu\geq 0$, $r\in\ C^1(\bar{\Omega}\times[0,\infty))\cap L^\infty\!\left(\Omega\times(0,\infty)\right)$ is nonnegative and the functions $f,g\in C^1\!\left([0,\infty)\right)$ are assumed to satisfy $f(0)\geq0$, $g(0)\geq0$ as well as $$-k_f s^{\alpha} -l_f\leq f(s)\leq -K_f s^\alpha+L_f\ \text{ and }\ -k_g s^\beta -l_g\leq g(s)\leq -K_g s^\beta+L_g\quad\text{for }s\geq0,$$ respectively, with constants $\alpha,\beta>1$, $k_f,K_f,k_g,K_g>0$ and $l_f,L_f,l_g,L_g\geq0$ and $\alpha,\beta>1$. Assuming that $\alpha>1+\sqrt{2}$, $\min\{\alpha,\beta\}>\frac{\alpha+1}{\alpha-1}$ and that $r$ satisfies certain structural conditions we establish the global solvability of this system with respect to a suitable generalized solution concept and then, for the more restrictive case of $\alpha,\beta>1+\sqrt{2}$ and $\mu>0$, investigate an eventual regularity effect driven by the decay of the nutrient density $w$.