Global weak solutions to a strongly degenerate haptotaxis model

TL;DR

This work analyzes a one-dimensional, strongly degenerate haptotaxis model coupling a degenerate diffusion–taxis equation for tumor density $u$ with an ODE for tissue density $w$, under a no-flux boundary condition. The authors introduce nondegenerate regularizations, derive an entropy-type estimate that yields key a priori bounds, and establish weak compactness for nonlinear terms via Dunford–Pettis arguments. They construct limit functions on the region where diffusion is positive and extend them to the degenerate boundary by solving a local ODE at each boundary point, ultimately proving the global existence of weak solutions to the original degenerate problem. The approach combines entropy methods, compactness (Aubin–Lions, Dunford–Pettis), and a careful boundary-patching argument to handle strong degeneracy and cross-diffusion effects, contributing to the mathematical understanding of degenerate taxis models in tissue networks.

Abstract

We consider a one-dimensional version of a model obtained in [C. Engwer, A. Hunt, and C. Surulescu: Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings, IMA J. Math. Med. Biol. (2015), doi: 10.1093/imammb/dqv030] and describing the anisotropic spread of tumor cells in a tissue network. The model consists of a reaction-diffusion-taxis equation for the density of tumor cells coupled with an ODE for the density of tissue fibers and allows for strong degeneracy both in the diffusion and the haptotaxis terms. In this setting we prove the global existence of weak solutions to an associated no-flux initial-boundary value problem.

Theorems & Definitions (21)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Corollary 3.4
• Lemma 3.5