Blow-up properties for a degenerate parabolic system coupled via nonlinear boundary flux
Tóm tắt
In this paper, we study the simultaneous and non-simultaneous blow-up problem for a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain hypotheses on the initial data and parameters, we prove that non-simultaneous blow-up is possible. Moreover, we get some conditions on which simultaneous blow-up must occur, as well as the non-simultaneous blow-up conditions for every initial data. Furthermore, we get a result on the coexistence of both simultaneous and non-simultaneous blow-ups.
Tài liệu tham khảo
Bebernes, J, Eberly, D: Mathematical Problem from Combustion Theory. Appl. Math. Sci., vol. 83. Springer, Berlin (1989)
Lieberman, GM: Second Parabolic Differential Equations. World Scientific, Singapore (1996)
Song, XF, Zheng, SN: Blow-up analysis for a quasilinear parabolic system with multi-coupled nonlinearities. J. Math. Anal. Appl. 281, 739-756 (2003)
Brändle, C, Quirós, F, Rossi, JD: Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun. Pure Appl. Anal. 4(3), 523-536 (2005)
Brändle, C, Quirós, F, Rossi, JD: The role of non-linear diffusion in non-simultaneous blow-up. J. Math. Anal. Appl. 308, 92-104 (2005)
Pinasco, JP, Rossi, JD: Simultaneous versus non-simultaneous blow-up. N.Z. J. Math. 29, 55-59 (2000)
Quirós, F, Rossi, JD: Non-simultaneous blow-up in a semilinear parabolic system. Z. Angew. Math. Phys. 52, 342-346 (2001)
Quirós, F, Rossi, JD: Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud. 3, 397-418 (2003)
Rossi, JD, Souplet, P: Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differ. Integral Equ. 18, 405-418 (2005)
Zheng, SN, Qiao, L: Non-simultaneous blow-up in a reaction-diffusion system. Appl. Comput. Math. 180, 309-317 (2006)
Zheng, SN, Liu, BC, Li, FJ: Non-simultaneous blow-up for a multi-coupled reaction-diffusion system. Nonlinear Anal. 64, 1189-1202 (2006)
Zheng, SN, Liu, BC, Li, FJ: Simultaneous and non-simultaneous blow-up for a cross-coupled parabolic system. J. Math. Anal. Appl. 326, 414-431 (2007)
Liu, BC, Li, FJ: Non-simultaneous blow-up and blow-up rates for reaction-diffusion equations. Nonlinear Anal. 13, 764-778 (2012)
Ling, ZQ, Wang, ZJ, Zhang, GQ: The simultaneous and non-simultaneous blow-up criteria for a diffusion system. Acta Math. Sci. 33(1), 139-149 (2013)
Jiang, ZX, Zheng, SN, Song, XF: Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions. Appl. Math. Lett. 17, 193-199 (2004)
Pablo, AD, Quirós, F, Rossi, JD: Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition. IMA J. Appl. Math. 67, 69-98 (2002)
Song, XF, Zheng, SN: Blow-up and blow-up rate for a reaction-diffusion model with multiple nonlinearities. Nonlinear Anal. 54, 279-289 (2003)
Jiang, ZX, Zheng, SN: Blow-up rate for a nonlinear diffusion equation with absorption and nonlinear boundary flux. Adv. Math. 33, 615-620 (2004)
Friedman, A: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)