Mathematical Analysis of an Acid Induced Cancer Invasion Model-the Existence of Traveling Wave

.In this work, a reaction-diffusion system modeling tumor invasion is investigated. The model is proposed by renowned biologist and cancer expert Gatenby and his collaborator [R. A. Gatenby and E. T. Gawlinski, Cancer Res., 56 (1996), pp. 5745–5753]. The underlying partial differential equation system has cross-diffusion and density-dependent degenerate diffusion coefficient and involves the interactions of three quantities: tumor tissue density, acid concentration, and normal tissue density. The most challenging and interesting issue for the analysis of the current model is the existence of the traveling wave solution which signifies the invasion of tumor tissue into normal tissue. Because the diffusion of tumor tissue is influenced by the density of normal cells (cross-diffusion) and the diffusion coefficient is zero when normal cells are at the carrying capacity, this causes the diffusion coefficient of tumor tissue to be degenerate, a hallmark of the underlying partial differential equation (PDE) system. A rigorous proof of existence of traveling wave is proved for one-dimensional case.Keywordstumor invasiondegenerate diffusioncross-diffusiontraveling waveexistenceMSC codes35A0135K5135K55