Systems with local and nonlocal diffusions, mixed boundary conditions, and reaction terms. (English) Zbl 1470.35381

Summary: We study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.


35R09 Integro-partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
35B44 Blow-up in context of PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
Full Text: DOI


[1] Andreu-Vaillo, F.; Mazón, J. M.; Rossi, J. .; Toledo-Melero, J. J., Nonlocal Diffusion Problems. Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165 (2010), Providence, RI, USA: American Mathematical Society, Real Sociedad Matematica EspaNola, Madrid, Providence, RI, USA · Zbl 1214.45002 · doi:10.1090/surv/165
[2] Bates, P. W.; Fife, P. C.; Ren, X.; Wang, X., Traveling waves in a convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 138, 2, 105-136 (1997) · Zbl 0889.45012 · doi:10.1007/s002050050037
[3] Bogoya, M., A nonlocal nonlinear diffusion equation in higher space dimensions, Journal of Mathematical Analysis and Applications, 344, 2, 601-615 (2008) · Zbl 1145.35009 · doi:10.1016/j.jmaa.2008.02.067
[4] Cortazar, C.; Elgueta, M.; Rossi, J. D., A nonlocal diffusion equation whose solutions develop a free boundary, Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics, 6, 2, 269-281 (2005) · Zbl 1067.35136 · doi:10.1007/s00023-005-0206-z
[5] Chasseigne, E.; Chaves, M.; Rossi, J. D., Asymptotic behavior for nonlocal diffusion equations, Journal de Mathématiques Pures et Appliquées, 86, 3, 271-291 (2006) · Zbl 1126.35081 · doi:10.1016/j.matpur.2006.04.005
[6] Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, 153-191 (2003), Berlin, Germany: Springer, Berlin, Germany · Zbl 1072.35005
[7] Garcia-Melian, J.; Rossi, J. D., On the principal eigenvalue of some nonlocal diffusion problems, Journal of Differential Equations, 246, 1, 21-38 (2009) · Zbl 1162.35055 · doi:10.1016/j.jde.2008.04.015
[8] Pérez-Llanos, M.; Rossi, J. D., Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, 70, 4, 1629-1640 (2009) · Zbl 1169.35312 · doi:10.1016/j.na.2008.02.076
[9] Bogoya, M., On non-local reaction-diffusion system in a bounded domain, Boundary Value Problems (2018) · Zbl 1499.35115 · doi:10.1186/s13661-018-0958-2
[10] Escobedo, M.; Herrero, M. A., A semilinear parabolic system in a bounded domain, Annali di Matematica Pura ed Applicata. Serie Quarta, 165, 315-336 (1993) · Zbl 0806.35088 · doi:10.1007/BF01765854
[11] Bandle, C.; Brunner, H., Blow-up in diffusion equations, Journal of Computational and Applied Mathematics, 97, 1-2, 3-22 (1998) · Zbl 0932.65098 · doi:10.1016/s0377-0427(98)00100-9
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