Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 55, pp. 1-17.

Title: Turing instability analysis of a singular cross-diffusion problem
       
Authors: Gonzalo Galiano (Univ. of Oviedo, Spain)
         Victor Gonzalez-Tabernero (Univ. of Santiago de Compostela, Spain)

Abstract: 
 The population model by Busenberg and Travis is a paradigmatic model in ecology
 and tumor modeling because its ability to capture interesting phenomena
 such as segregation of populations. Its singular mathematical structure enforces
 the consideration of regularized problems to deduce properties as fundamental
 as the existence of solutions.
 In this article we perform a weakly nonlinear stability analysis of a general
 class of regularized problems to study the convergence of the instability modes
 in the limit of the regularization parameter. We demonstrate with some specific
 examples that the pattern formation observed in the regularized problems,
 with unbounded wave numbers, is not present in the limit problem because of the
 amplitude decay of the oscillations.
 We also check the results of the stability analysis with direct finite element
 simulations of the problem.

Submitted December 8, 2020. Published June 21, 2021.
Math Subject Classifications: 35K55, 35B36, 92D25.
Key Words: Cross-diffusion; Turing instability; weakly nonlinear equation;
           finite element method.