The traveling wave to a reaction-diffusion system modeling cancer invasion

In this paper, we study the existence of traveling wave to a cross-diffusion and degenerate reaction-diffusion system. The underlying biological problem is to describe the process in microscopic scale the invasion of cancer cells to the damage of healthy tissues. The mathematical model is established by renowned biologist and cancer expert Gatenby and his collaborator Gawlinski [10] , and later generalized in [19] to cover wide range of parameter setting and more intricate interactions between the underlying mathematical quantities which represent tumor density, acid concentration and healthy tissue density. Mathematically, the cancer invasion phenomenon is quantitatively depicted by the existence of a traveling wave solution connecting two equilibrium states, one of which is the state of normal tissue without cancer cells, and the other with presence of cancer cells, and normal tissue being either partially or completely eliminated (depending on the range of parameters). We prove the existence of traveling wave for all speed bigger or equal to some value, for a rather general setting of parameters to cover both cases of either normal tissue is partially or completely eliminated after cancer invasion.