Abstract
AbstractWe prove that, under a suitable rescaling of the integrable kernel defining the nonlocal diffusion terms, the corresponding sequence of solutions of the Shigesada–Kawasaki–Teramoto nonlocal cross-diffusion problem converges to a solution of the usual problem with local diffusion. In particular, the result may be regarded as a new proof of existence of solutions for the local diffusion problem.
Funder
Ministerio de Economía, Industria y Competitividad, Gobierno de España
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Geometry and Topology,Algebra and Number Theory,Analysis
Reference22 articles.
1. Adams, R.A.: Sobolev Spaces. Academic Press (1975)
2. Amann, H.., III., Dynamic theory of quasilinear parabolic systems: Global existence. Math. Z. 202, 219–250 (1989)
3. Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo-Melero, J.J.: Nonlocal diffusion problems. American Mathematical Society (2010)
4. Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Menaldi, J.L., (eds.) Optimal Control and Partial Differential Equations, 439–455, IOS Press (2001)
5. Chen, L., Jüngel, A.: Analysis of a multidimensional parabolic population model with strong cross-diffusion. SIAM J. Math. Anal. 36(1), 301–322 (2004)