Haptotaxis–Chemotaxis Tumor Invasion with Weighted Mass-Conservative Quantity

Abstract

We consider a haptotaxis–chemotaxis tumor invasion model, which is derived by mixing two models. One model is proposed by Chaplain and Anderson in 2002 in which the haptotaxis effect is considered. The other has a conservative quantity and is proposed by Fujie, Yokota, and the author in 2014 in which the chemotaxis effect is considered. In addition, the coefficient of random motility of tumor cells is degenerate and depends on chemoattractant substance, extracellular matrix, and matrix degrading enzymes, which are unknown in the model. This situation gives quasi-variational structures to not only time-dependent subdifferentials of convex functions but also inner products on a real Hilbert space $$(H^1(\Omega ))^*$$ . In order to overcome the mathematical difficulty, which comes from such quasi-variational structures and mass-conservative quantity, we apply the general theory of quasi-variational evolution inclusions with conservative properties on real Hilbert spaces which is established by the author in 2023 and show the existence of strong solutions to the tumor invasion model under consideration.