Global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread

In this paper, we consider the following system { u t = Δ u − χ ∇ ⋅ ( u ∇ ω ) , v t = d v Δ v − ξ ∇ ⋅ ( v ∇ ω ) , m t = d m Δ m + u − m , ω t = − ( γ 1 u + m ) ω , in two dimensional space with zero-flux boundary conditions. This model was proposed by Franssen et al. [7] to characterize the invasion and metastatic spread of cancer cells. We first establish the global existence of uniformly bounded global strong solutions. Then using the decay of ECM and the positivity of MDE, we further improve the regularity of obtained solutions, and achieve the uniform boundedness of solutions in the classical sense. Subsequently, we also prove the uniqueness of solutions. After that, we turn our attention to the large time behavior of solutions, and show that the global classical solution strongly converges to a semi-trivial steady state in the large time limit.