Global boundedness and asymptotic behavior in a double haptotaxis model for oncolytic virotherapy

This work studies a haptotactic cross-diffusion system modeling oncolytic virotherapy{ut=Δu−∇⋅(u∇v)−ρuz,vt=−(u+w)v−κv,wt=DwΔw−ξw∇⋅(w∇v)−w+ρuz,zt=DzΔz−z−ρuz+βw, in a smooth bounded domain Ω⊂R2, with parameters Dw>0, Dz>0, ξw≥0, ρ≥0, κ>0 and β≥0. This system describes the interaction among uninfected and infected cancer cells, extracellular matrix and oncolytic viruses. It is rigorously proved that an associated no-flux initial-boundary value problem has a unique global classical solution which is bounded in (L∞(Ω))4. Moreover, it is shown that this bounded solution can approach a spatially constant equilibrium in the large time limit under the additional assumption that 0≤β