# Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis-haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)

## TL;DR

This paper introduces a class of chemotaxis-haptotaxis models with indirect signal production, proving global classical solvability in 2D and showing that indirectness suppresses blow-up phenomena.

## Contribution

It presents the first rigorous proof of global classical solutions for a broad class of indirect signal production chemotaxis models in two dimensions.

## Key findings

- Global classical solutions exist in 2D under mild conditions.
- Indirect signal production reduces blow-up potential in chemotaxis systems.
- The model applies to cancer invasion and other biological processes.

## Abstract

We propose and study a class of parabolic-ODE models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumor microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature consituting a core difference to both classical Keller-Segel chemotaxis systems and Chaplain-Lolas type chemotaxis-haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.