Geodesic
On a problem of cross-diffusion with nonlocal boundary conditions
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 5, pp. 614-620.

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Condition of global existence of solution of a non-linear system of cross-diffusion with non-linear boundary conditions is studied in the paper. Critical exponents of Fujita type and critical exponents of global existence of solution are established.
Keywords: nonlinear parabolic system, critical Fujita curve, blow-up.
Mots-clés : cross-diffusion 
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Zafar R. Rakhmonov; Jasur E. Urunbayev. On a problem of cross-diffusion with nonlocal boundary conditions. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 5, pp. 614-620. http://geodesic.mathdoc.fr/item/JSFU_2019_12_5_a9/

[1] Z.Q. Wu, J.N. Zhao, J.X. Yin, H.L. Li, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001 | Zbl

[2] M.M. Aripov, Methods of Reference Equations for Solving Nonlinear Boundary Value Problems, Fan, Tashkent, 1988 | MR

[3] A.S. Kalashnikov, “Some Questions of the Qualitative Theory of Nonlinear Degenerate Second-order Parabolic Equations”, Uspekhi Mat. Nauk, 42:2 (254) (1987), 135–176 | MR | Zbl

[4] J.D. Murray, Mathematical Biology, 3rd ed., Springer, Berlin, 2002 | MR | Zbl

[5] H. Malchow, S.V. Petrovskii, E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations, Chapman Hall/CRC Press, London, 2008 | MR

[6] M.A. Tsyganov, V.N. Biktashev, J. Brindley, A.V. Holden, G.R. Ivanitsky, “Waves in Cross-diffusion Systems — a Special Class of Nonlinear Waves”, UFN, 177:3 (2007), 275–300 | DOI

[7] H. Levine, “The Role of Critical Exponents in Blowup Theorems”, SIAM Rev., 32:2 (1990), 262–288 | DOI | MR | Zbl

[8] S. Wang, C.H. Xie, M.X. Wang, “Note on Critical Exponents for a System of Heat Equations Coupled in the Boundary Conditions”, J. Math. Analysis Applic., 218 (1998), 313–324 | DOI | MR | Zbl

[9] S. Wang, C.H. Xie, M.X. Wang, “The Blow-up Rate for a System of Heat Equations Completely Coupled in the Boundary Conditions”, Nonlinear Anal., 35 (1999), 389–398 | DOI | MR | Zbl

[10] F. Quiros, J.D. Rossi, “Blow-up Set and Fujita-type Curves for a Degenerate Parabolic System with Nonlinear Conditions”, Indiana Univ. Math. J., 50 (2001), 629–654 | DOI | MR | Zbl

[11] S.N. Zheng, X.F. Song, Z.X. Jiang, “Critical Fujita Exponents for Degenerate Parabolic Equations Coupled Via Nonlinear Boundary Flux”, J. Math. Anal. Appl., 298 (2004), 308–324 | DOI | MR | Zbl

[12] Z. Rakhmonov, “On the Properties of Solutions of Multidimensional Nonlinear Filtration Problem with Variable Density and Nonlocal Boundary Condition in the Case of Fast Diffusion”, J. Sib. Fed. Univ. Math. $\$ Phys., 9:2 (2016), 236–245

[13] Z. Rakhmonov, “Assessment for the Solutions of a Nonlinear System of Heat Conduction Equations with Variable Density and Nonlocal Boundary Condition”, Bulletin NUU, 2016, no. 1(2), 145–154 | MR

[14] M.M. Aripov, A.S. Matyakubov, “To the Qualitative Properties of Solution of System Equations not in Divergence Form of Polytrophic Filtration in Variable Density”, Nanosystems: Physics, Chemistry, Mathematics, 8:3 (2017), 317–322 | DOI

[15] M.M. Aripov, A.S. Matyakubov, “Self-similar Solutions of a Cross-diffusion Parabolic System with Variable Density: Explicit Estimates and Asymptotic Behavior”, Nanosystems: Physics, Chemistry, Mathematics, 8:1 (2017), 5–12 | DOI | MR

[16] M.M. Aripov, Z.R. Rakhmonov, “To the asymptotics of the solutions of a nonlinear heat conduction problem with a gradient nonlinearity”, Uzbek Math. J., 2013, no. 3, 19–27 | MR