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From non-local to classical SKT systems: triangular case with bounded coefficients
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This paper solves partially a question suggested by Fontbona and M\'el\'eard in a paper published in 2015. The issue is to obtain rigorously cross-diffusion systems \`a la Shigesada-Kawasaki-Teramoto as the limit of relaxed systems in which the cross-diffusion and reaction coefficients are non-local. We depart from the existence result established by Fontbona M\'el\'eard for a general class of non-local systems and study the corresponding asymptotic as the convolution kernels tend to Dirac masses, but only in the case of (strictly) triangular systems, with bounded coefficients. Our approach is based on a new result of compactness for the Kolmogorov equation, which is reminiscent of the celebrated duality lemma of Michel Pierre.
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Refined blow-up criteria and global solutions for triangular cross-diffusion systems
Establishes refined L^p-based blow-up criteria for triangular SKT cross-diffusion systems via hierarchical structure and tame Sobolev estimates, and proves global existence of non-negative strong solutions for two-spe...
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