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arXiv · 2607.01857

Refined blow-up criteria and global solutions for triangular cross-diffusion systems

Alexandre Bertolino
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Abstract

We study the Cauchy problem associated with a class of triangular cross-diffusion systems of Shigesada-Kawasaki-Teramoto type. We develop a self-contained well-posedness theory in C 0 ([0, T ]; H s (T d )) based on regularity estimates for scalar Kolmogorov equations. The diffusion coefficient of each species depends only on species of lower index, yielding a hierarchical structure that allows for refined blow-up criteria. Finite-time singularities can occur only through the divergence of the L $\infty$ (T d ) norm of the solution. Assuming polynomial growth of the nonlinearities, this criterion is refined to an L p -based blow-up condition for some finite exponent p, yielding a substantially weaker obstruction to global existence than classical Sobolev blow-up criteria. The proof is achieved through refined tame estimates for composition in Sobolev spaces. As an application, we prove global existence of non-negative strong solutions for two-species systems with logistic-type reaction terms in dimensions d $\le$ 2.

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Alexandre Bertolino. 2026-07-02. Refined blow-up criteria and global solutions for triangular cross-diffusion systems. https://arxiv.org/abs/2607.01857

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