Global weak solutions in a three-dimensional two-species cancer invasion haptotaxis model without cell proliferation

Abstract

This paper considers the two species cancer invasion haptotaxis model without cell proliferation in three space dimensions. The system consists of two parabolic partial differential equations (PDEs) describing the migration of differentiated cancer cells and cancer stem cells and the epithelial–mesenchymal transition between the two families of cells, a parabolic/elliptic PDE governing the evolution of matrix degrading enzymes, and an ordinary differential equation reflecting the degradation and remodeling of the extracellular matrix. We underline that the absence of a logistic source aggravates mathematical difficulties that are overcome by constructing a delicate energy-functional. For any suitably regular initial data, we establish the global existence of weak solutions to the associated initial-boundary value problem. This result affirmatively answers the open question proposed by Dai and Liu [SIAM J. Math. Anal. 54, 1–35 (2022)].