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

Abstract

This study considers a model for oncolytic virotherapy, as given by the reaction–diffusion–taxis system \[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v, \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*}\] in a smoothly bounded domain Ω ⊂ ℝ 2 , with parameters D w > 0, D z > 0, β > 0 and ρ ⩾ 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 $({1}/{|\Omega |})\int _\Omega u(\cdot ,0) > 1/(\beta -1)$ , infinite-time blow-up occurs at least in the particular case when ρ = 0. In order to provide an appropriate complement to this, the current study reveals that for any ρ ⩾ 0 and arbitrary β > 0, at each prescribed level γ ∈ (0, 1/( β − 1) + ) one can identify an L ∞ -neighbourhood of the homogeneous distribution ( u , v , w , z ) ≡ (γ, 0, 0, 0) within which all initial data lead to globally bounded solutions that stabilize towards the constant equilibrium ( u ∞ , 0, 0, 0) with some u ∞ > 0.

Topics & Concepts