Boundedness in a quasilinear forager–exploiter model.

We study a forager–exploiter model with nonlinear diffusions: ut=∇·(D(u)∇u)−∇·(u∇w),x∈Ω,t>0,vt=∇·(D¯(v)∇v)−∇·(v∇u),x∈Ω,t>0,wt=Δw−(u+v)w−w+r(x,t),x∈Ω,t>0,$$\begin{equation*} \hspace*{58pt}{\left\lbrace \def\eqcellsep{&}\begin{array}{lll} u_t=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (u\nabla w),&x\in \Omega,\ \ t>0,\\[3pt] v_t=\nabla \cdot (\overline{D}(v)\nabla v)-\nabla \cdot (v\nabla u),&x\in \Omega,\ \ t>0,\\[3pt] w_t=\Delta w-(u+v)w- w+r(x,t),\; &x\in \Omega,\ \ t>0, \end{array} \right.} \end{equation*}$$where Ω⊂R3$\Omega \subset \mathbb {R}^3$ is a smooth bounded domain, r$r$ is a nonnegative bounded function, D,D¯∈C2([0,∞))$D,\overline{D}\in C^2([0,\infty))$ satisfying D,D¯>0in[0,∞),lim infs→∞D¯(s)s>0,lim infs→∞D(s)>K$$\begin{equation*} \hspace*{52pt}D,\overline{D}>0\,\,\text{in}\,\,[0,\infty), \ \ \liminf _{s\rightarrow \infty }\displaystyle \frac{\overline{D}(s)}{s}>0,\ \ \liminf _{s\rightarrow \infty }D(s)>K \end{equation*}$$with some sufficiently large K>0$K>0$. Global‐in‐time solutions are established for corresponding Neumann initial‐boundary value problem. [ABSTRACT FROM AUTHOR]