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Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 1 / February 2022
- Published online by Cambridge University Press:
- 08 January 2021, pp. 81-101
- Print publication:
- February 2022
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- Article
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This study considers a model for oncolytic virotherapy, as given by the reaction–diffusion–taxis system
in a smoothly bounded domain Ω ⊂ ℝ2, with parameters Dw > 0, Dz > 0, β > 0 and ρ ⩾ 0.
\[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v, \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*}\]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.
