Abstract 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.

中文翻译：

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全局经典解和收敛到癌细胞侵袭和转移扩散的数学模型

摘要 在本文中，我们考虑以下系统 { ut = Δ u − χ ∇ ⋅ ( u ∇ ω ) , vt = dv Δ v − ξ ∇ ⋅ ( v ∇ ω ) , mt = dm Δ m + u − m , ω t = − ( γ 1 u + m ) ω ，在具有零通量边界条件的二维空间中。该模型由 Franssen 等人提出。[7] 表征癌细胞的侵袭和转移扩散。我们首先建立一致有界全局强解的全局存在性。然后利用ECM的衰减和MDE的正性，进一步提高所得解的正则性，实现经典意义上解的均匀有界。随后，我们也证明了解的唯一性。之后，我们将注意力转向解决方案的长时间行为，