The paper considers the problem \begin{align*} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \mu u(1-u) - uz, \\ v_t = -(u+w) v, \\ w_t = \Delta w - \nabla \cdot (w \nabla v) - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{cases} \end{align*} complemented with initial and, for the PDEs, with no-flux boundary conditions, in smooth, bounded domains \(\Omega \subset \mathbb R^3\). Such a system arises in oncolytic virotherapy. More concretely, \(u\), \(v\), \(w\) and \(z\) denote (densities of) uninfected cancer cells, the extracellular matrix, infected cancer cells and the oncolytic virus, respectively. In contrast to several precedents, the paper under consideration accounts for haptotactic movement also of the infected cells, i.e., the term \(-\nabla \cdot (w \nabla v)\) is included in the system. (However, as it is often done in the literature, the authors neglect the remodelling term \(+\mu_v(1-v)\) of the extracellular matrix which is present in the model originally introduced in [T. Alzahrani et al., Math. Biosci. 310, 76–95 (2019; Zbl 1425.92103)].)
The main result states that if the initial data and \(\beta\) are suitably small (although the latter may be larger than \(1\)), then there exists a global classical solution which converges to the homogeneous equilibrium \((1, 0, 0, 0)\).
One key idea of the proof is to consider the largest interval \((0, T)\) in which \(\|z(\cdot, t)\|_{L^\infty(\Omega)} \le \sqrt{\varepsilon} \mathrm{e}^{-\delta t}\) (for suitably chosen \(\varepsilon\) and \(\delta\)). Working in \((0, T)\) then allows the authors to favourably apply the comparison principle for cooperative parabolic systems to certain transformed quantities of \(u\) and \(w\), which constitutes a key step towards showing that \(T = T_{\max}\) if \(T_{\max} < \infty\).