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This paper considers a model for oncolytic virotherapy given by a thiply haptotactic cross-diffusion system
ut = DuΔu − ξu∇ · (u∇v) + μuu(1 − u) − ρuuz, x ∈ Ω, t > 0,
vt = −(αuu + αww)v + μvv(1 − v), x ∈ Ω, t > 0,
wt = DwΔw − ξw∇ · (w∇v) − δww + ρwuz, x ∈ Ω, t > 0,
zt = DzΔz − ξz∇ · (z∇v) − δzz − ρzuz + βw, x ∈ Ω, t > 0,
(Du∇u − ξuu∇v) · ν = (Dw∇w − ξww∇v) · ν = (Dz∇z − ξzz∇v) · ν = 0, x ∈ ∂Ω, t > 0,
u(x, 0) = u0(x), v(x, 0) = v0(x), w(x, 0) = w0(x), z(x, 0) = z0(x), x ∈ Ω,
(0.1)
in a bounded domain Ω ⊂ RN (N = 1, 2) with smooth boundary. We prove that when N = 1, the system possesses a unique global bounded solution. When N = 2, if there exists some constant K > 0 such that Dz > ξzK, then the solution is global in time. And if we add the condition δwρz > ρwβ > 0, the solution is globally bounded. Furthermore, with some suitable assumption, we could also obtain that the solution exponentially stabilizes to the constant equilibrium (1,0,0,0) as t → ∞.