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 \Delta u - \chi_1 \nabla \cdot (u \nabla v) + u(\lambda_1 - \mu_1 u + a_1 v) \ v_t = D_2 \Delta v + \chi_2 \nabla \cdot (v \nabla u) + v(\lambda_2 - \mu_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 $\varepsilon > 0$ 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}(\Omega)} + |v_0 - v\star|{W^{2,2}(\Omega)} < \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.