This talk will present the work carried out as part of my PhD thesis with Maxime Breden. Multi-species diffusion reaction equations are systems with singular solving difficulties for each system. However, thanks to computer-assisted methods, it is possible to establish properties of these equations in a systematic way.

 

We will begin by looking at a chemotaxis system with cross-diffusion. The specificity of this system is the management of non-polynomial non-linearities. After establishing a suitable solution space and a “toolbox” based on Neumann series and Taylor expansions, we use a Newton-Kantorovitch type theorem to show the existence of many stationary states. We then look at a diffusion reaction system. We will establish the stability of the system with respect to a free parameter, with a first method using Gershgorin disks and then with a second method again using a Newton-Kantorovitch theorem.