A Self-Similar Analysis of the Solutions to the Cross-Diffusion System

We study the qualitative behaviour of weak solutions to a doubly nonlinear cross-diffusion system in an inhomogeneous medium with nonlinear boundary flux. The main novelty of the paper lies in the analysis of a cross-diffusive p-Laplacian system weighted by a spatially varying density in the form ρ(x)=(1+|x|)n, combined with nonlinear boundary interactions. By constructing self-similar weak solutions of Barenblatt type and employing a nonlinear separation of variables, the system is reduced to a coupled family of ordinary differential equations that characterise admissible similarity profiles. This approach allows us to identify Fujita-type critical conditions separating global solutions from finite-time blow-up and to derive explicit estimates for the solution profiles. Consequently, we establish critical thresholds driven by the interaction between cross-diffusion, degeneracy, boundary nonlinearity, and medium inhomogeneity.