Global existence of a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant

This paper deals with the cancer invasion model {u_t=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,v_t=Δv−v+u,x∈Ω,t>0,w_t=−vw+ηw(1−w−u),x∈Ω,t>0 in a bounded smooth domain Ω⊂R² with zero-flux boundary conditions, where χ,ξ, μ and η are positive parameters. Compared to previous mathematical studies, the novelty here lies in: first, our treatment of the full parabolic chemotaxis–haptotaxis system; and second, allowing for positive values of η, reflecting processes with self-remodeling of the extracellular matrix. Under appropriate regularity assumptions on the initial data (u₀,v₀,w₀), by using adapted L^p-estimate techniques, we prove the global existence and uniqueness of classical solutions when μ is sufficiently large, i.e., in the high cell proliferation rate regime.