Asymptotic stability of spatial homogeneity in a haptotxis model for oncolytic virotherapy

Abstract

This work considers a model for oncolytic virotherapy, as given by the reaction-diffusion-taxis system $$\{\begin{array}{l}{u}_t=\Delta u-\nabla \cdot (u\nabla v)-\rho uz,\\ v_t=-(u+w)v,\\ w_t=D_w\Delta w-w+uz,\\ z_t=D_z\Delta z-z-uz+\beta w,\end{array}$$ in a smoothly bounded domain $\Omega \subset {\mathbb{R}}^2$, with parameters $D_w>0,D_z>0,\beta >0$ and $\rho \ge 0$.\\ % Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if $\beta <1$, whereas whenever $\beta >1$ and $\frac{1}{|\Omega |}{\int }_{\Omega }u(\cdot ,0)>\frac{1}{\beta -1}$, infinite-time blow-up occurs at least in the particular case when $\rho =0$.\abs % In order to provide an appropriate complement to this, the present work reveals that for any $\rho \ge 0$ and arbitrary $\beta >0$, at each prescribed level $\gamma \in (0,\frac{1}{(\beta -1{)}_{+}})$ one can identify an $L^{\infty }$-neighborhood of the homogeneous distribution $(u,v,w,z)\equiv (\gamma ,0,0,0)$ within which all initial data lead to globally bounded solutions that stabilize toward the constant equilibrium $(u_{\infty },0,0,0)$ with some $u_{\infty }>0$.

21 pages