Communications in Mathematical Sciences

Volume 15 (2017)

Number 6

Global weak solutions to a strongly degenerate haptotaxis model

Pages: 1581 – 1616



Michael Winkler (Institut für Mathematik, Universität Paderborn, Germany)

Christina Surulescu (Felix-Klein-Zentrum für Mathematik, Technische Universität Kaiserslautern, Germany)


We consider a one-dimensional version of a model obtained in [C. Engwer, A. Hunt, and C. Surulescu, IMA J. Math. Med. Biol., 33(4):435–459, 2016] 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. Numerical simulations are performed in order to illustrate the model behavior.


haptotaxis, degenerate diffusion, global existence

2010 Mathematics Subject Classification

Primary 35K51, 35K57, 35K65. Secondary 35D30, 35K55, 35Q30, 35Q92, 92C17.

Received 23 June 2016

Published 27 June 2017