Mathematical analysis of a tumor invasion model – global existence of weak solutions

A no-flux initial-boundary value problem modeling tumor invasionequation* cases uₜ = ∇ · ((1-w)∇ u)+ u(1-u), \\ vₜ = Δ v -v +u, \\ wₜ = -vw + μ w(1-w) cases equation*is studied in a bounded smooth domain ΩN (N≥ 1), where u, v, and w represent the densities of neoplastic tissue, H⁺ ions, and normal tissue, respectively. The most distinguished feature of this model is that the diffusion of tumor cells is influenced by the density of normal cells, and diffusion degeneracy arises when normal cells are at their carrying capacity. In our previous paper [19], we proved the global existence and stability of classical solutions under the assumption that the initial density of normal cells satisfies w(·, 0)<1. This assumption guarantees that normal cells are below their carrying capacity and no diffusion degeneracy appears in the system, hence classical solutions can be derived. In this work, we focus on a more general case when w(·, 0)≤ 1, that is, the density of normal cells may reach their carrying capacity at some subset of the domain Ω. Compared to [19], in this paper we develop some new techniques and present a rigorous proof of the global existence of weak solutions under this much weaker assumption for w(·, 0).