A Result on Moser Iteration in Haptotaxis Systems and Application to Cancer Invasion Models with Tissue Reconstruction

Source abstract

The manuscript first considers a general system of parabolic inequalities that contain a quadratic zero-order nonlinearity as a core constituent. Under mild assumptions inter alia requiring a certain non-degeneracy hypothesis, by means of a Moser-type iterative argument an $L^∞$ estimate for nonnegative solutions is derived from presupposed $L^p$ bounds with suitably large $p$. This general result is subsequently applied to an apparently novel type of haptotaxis cancer invasion models involving tissue regeneration by the cancer-associated fibroblast cells, as well as to a classical haptotaxis system modeling tumor evolution in the presence of tissue self-remodeling. In both cases, global classical solutions to associated boundary value problems in domains of arbitrary dimension are thereby obtained under explicit assumptions on the strength of logistic degradation.