Finite-time blow-up in the higher dimensional parabolic-parabolic-ODE minimal chemotaxis-haptotaxis system

In this article, we consider the following parabolic-parabolic-ODE minimal chemotaxis-haptotaxis system {u<inf>t</inf>=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w),x∈Ω, t>0,v<inf>t</inf>=Δv−v+u,x∈Ω, t>0,w<inf>t</inf>=−vw,x∈Ω, t>0,[Formula presented]=0,x∈∂Ω, t>0,u(x,0)=u<inf>0</inf>(x),v(x,0)=v<inf>0</inf>(x),w(x,0)=w<inf>0</inf>(x),x∈Ω, in a bounded domain Ω⊂Rⁿ,n⩾3 with smooth boundary. We show that the finite time blow-up occurs to the above system. More specifically, we establish that in a radial setting, the generic mass blow-up phenomenon observed in corresponding chemotaxis-only systems (obtained by setting w≡0) is preserved in chemotaxis-haptotaxis system. Our proof demonstrates that for a given initial mass, there exists radially symmetric positive initial data for which the corresponding solution blows-up. Moreover, we illustrate that such initial data constitute a considerably large set in the sense that it is dense in C⁰(Ω‾)×W^1,∞(Ω)×C²(Ω‾) with respect to topology in L^p(Ω)×W^1,2(Ω)×L^∞(Ω), with p∈(1,[Formula presented]). Analogous results for corresponding parabolic-elliptic-ODE system [53] are already available, where the parabolic-parabolic-ODE case was open for further study.