Asymptotic behavior for solutions to an oncolytic virotherapy model involving triply haptotactic terms

In this paper, based on L^p- L^q estimate for the Neumann heat semigroup, we investigate the asymptotic behavior for solutions to an oncolytic virotherapy model given by {ut=Δu-ξu∇·(u∇v)-ρuuz,x∈Ω,t>0,wt=Δw-ξw∇·(w∇v)-δww+ρwuz,x∈Ω,t>0,vt=-(αuu+αww)v-δvv,x∈Ω,t>0,zt=Δz-ξz∇·(z∇v)-δzz-ρzuz+βw,x∈Ω,t>0,where u, w, v and z denote the density of uninfected cancer cells, oncolytic viruses infected cancer cells, extracellular matrix and oncolytic virus particles, respectively. It is showed that when suitably regular initial data satisfy a certain small condition, infected cancer cells and virus particle populations will both become extinct asymptotically.