Dampening effect of logistic source in a two-dimensional haptotaxis system with nonlinear zero-order interaction

TL;DR

This paper studies a complex oncolytic virotherapy model in two dimensions, proving global boundedness of solutions when logistic growth is present and showing stabilization to equilibrium under certain conditions.

Contribution

It extends previous results by establishing uniform boundedness and stability of solutions for the model with logistic source and nonlinear interactions.

Findings

Solutions are globally bounded when $$.

Solutions stabilize to the equilibrium $(1, 0, 0, 0)$ if $eta<1$.

The model's solutions converge in $L^p$ and $L^ty$ norms over time.

Abstract

This paper deals with the oncolytic virotherapy model \begin{equation}\begin{split} \begin{cases} &u_t = \Delta u - \nabla \cdot (u\nabla v)-uz +\mu u(1-u),& \\[2ex] &v_t = - (u+w)v,& \\[2ex] &w_t = D_w \Delta w - w + uz,& \\[2ex] &z_t = D_z \Delta z - z - uz + \beta w,& \end{cases} \end{split}\end{equation} in a bounded domain $\Omega$ $\subset$ $\Bbb{R}^2$ with smooth boundary, where $\mu$, $D_w$, $D_z$ and $\beta$ are prescribed positive parameters. For any given suitably regular initial data, the global existence of classical solution to the corresponding homogeneous Neumann initial-boundary problem for a more general model allowing $\mu=0$ was previously verified in $[$Y. Tao $\&$ M. Winkler, J. Differential Equations $\mathbf{268}$ (2020), 4973-4997$]$. This work further shows that whenever $\mu>0$, the above-mentioned global classical solution to the above equation is uniformly bounded; and moreover, if $\beta<1$, then the solution $(u, v, w, z)$ stabilizes to the constant equilibrium $(1, 0, 0, 0)$ in the topology $L^p(\Omega)\times (L^\infty(\Omega))^3$ with any $p>1$ in a large time limit.