Abstract: Tumor metastasis is an important link in the process of tumor development, and it is also one of the main reasons for cancer deterioration and treatment failure. Taking tumor metastasis as the background, a study is conducted on the generative model of tumor lymphatics based on the interaction between tumor and extracellular matrix (ECM). First, mathematical language is used to sort out the biological principles of tumor lymphangiogenesis, and then assumptions made and mathematical models established and qualitative analysis carried out. The proof of the uniqueness of the existence of local solutions of the model is mainly carried out by means of approximation methods, the qualitative theory of partial differential equations and Banach's immovable point theorem, as well as the uniqueness of the existence of the overall solution of the model with the help of the regularity estimate of the local solution and the embedding inequality. Finally, the difference numerical method is used to carry out numerical simulation to illustrate the reliability and accuracy of the model. This research is of great significance for in-depth understanding the mechanism of tumor metastasis, guiding cancer treatment, and promoting related research.

Key words: tumor lymphangiogenesis, extracellular matrix(ECM), reaction diffusion, existence, uniqueness