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

This work considers a model for oncolytic virotherapy, as given by the reaction-diffusion-taxis system $$ \left\{ \begin{array}{l} u_t = Δu - \nabla \cdot (u\nabla v)-ρuz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = D_w Δw - w + uz, \\[1mm] z_t = D_z Δz - z - uz + βw, \end{array} \right. $$ in a smoothly bounded domain $Ω\subset\mathbb{R}^2$, with parameters $D_w>0, D_z>0, β>0$ and $ρ\ge 0$.\\ % 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 $β<1$, whereas whenever $β>1$ and $\frac{1}{|Ω|}\int_Ω u(\cdot,0)>\frac{1}{β-1}$, infinite-time blow-up occurs at least in the particular case when $ρ=0$.\abs % In order to provide an appropriate complement to this, the present work reveals that for any $ρ\ge 0$ and arbitrary $β>0$, at each prescribed level $γ\in (0,\frac{1}{(β-1)_+})$ one can identify an $L^\infty$-neighborhood of the homogeneous distribution $(u,v,w,z)\equiv (γ,0,0,0)$ within which all initial data lead to globally bounded solutions that stabilize toward the constant equilibrium $(u_\infty,0,0,0)$ with some $u_\infty>0$.