Global solutions near homogeneous steady states in a multi-dimensional population model with both predator- and prey-taxis

Abstract

We study the system \begin{align*}\label{prob:star} \tag{$\star$} \begin{cases} u_t = D_1 Δu - χ_1 \nabla \cdot (u \nabla v) + u(λ_1 - μ_1 u + a_1 v) \\ v_t = D_2 Δv + χ_2 \nabla \cdot (v \nabla u) + v(λ_2 - μ_2 v - a_2 u) \end{cases} \end{align*} (inter alia) for $D_1,D_2,{\chi }_1,{\chi }_2,{\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2,a_1,a_2>0$ in smooth, bounded domains $\Omega \subset {\mathbb{R}}^n$, $n\in \{1,2,3\}$. Without any further restrictions on these parameters, we prove that there exists a constant stable steady state $(u_{\star },v_{\star })\in [0,\infty {)}^2$, meaning that there is $\epsilon >0$ such that, if $u_0,v_0\in W^{2,2}(\Omega )$ are nonnegative with ${\partial }_{\nu }{u}_0={\partial }_{\nu }{v}_0=0$ in the sense of traces and \begin{align*} \|u_0 - u_\star\|_{W^{2,2}(Ω)} + \|v_0 - v_\star\|_{W^{2,2}(Ω)} < \varepsilon, \end{align*} then there exists a global classical solution $(u,v)$ of \eqref{prob:star} with initial data $u_0,v_0$ converging to $(u_{\star },v_{\star })$ in $W^{2,2}(\Omega )$. Moreover, the convergence rate is exponential, except for the case ${\lambda }_2{\mu }_1={\lambda }_1{a}_2$, where it is is only algebraical.

27 pages; changed title and incorporated referee comments