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Properties of non-simultaneous blow-up solutions in nonlocal parabolic equations. (English) Zbl 1182.35053
Summary: This paper deals with blow-up solutions in parabolic equations coupled via nonlocal nonlinearities, subject to homogeneous Dirichlet conditions. Firstly, some criteria on non-simultaneous and simultaneous blow-up are given, including four kinds of phenomena: (i) the existence of non-simultaneous blow-up; (ii) the coexistence of non-simultaneous and simultaneous blow-up; (iii) any blow-up must be simultaneous; (iv) any blow-up must be non-simultaneous. Next, total versus single point blow-up are classified completely. Moreover, blow-up rates are obtained for both non-simultaneous and simultaneous blow-up solutions.

35B44 Blow-up in context of PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
35K59 Quasilinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B33 Critical exponents in context of PDEs
Full Text: DOI
[1] Escobedo, M.; Herrero, M.A., A semilinear parabolic system in a bounded domain, Ann. mat. pura appl., CLXV, 315-336, (1993) · Zbl 0806.35088
[2] Souplet, Ph., Blow-up in nonlocal reaction – diffusion equations, SIAM J. math. anal., 29, 1301-1334, (1998) · Zbl 0909.35073
[3] Chen, H.W., Global existence and blow-up for a nonlinear reaction – diffusion system, J. math. anal. appl., 212, 481-492, (1997) · Zbl 0884.35068
[4] Quirós, F.; Rossi, J.D., Non-simultaneous blow-up in a semilinear parabolic system, Z. angew. math. phys., 52, 342-346, (2001) · Zbl 0990.35057
[5] Brändle, C.; Quirós, F.; Rossi, J.D., The role of non-linear diffusion in non-simultaneous blow-up, J. math. anal. appl., 308, 92-104, (2005) · Zbl 1078.35045
[6] Li, F.J.; Liu, B.C.; Zheng, S.N., Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary flux, Z. angew. math. phys., 58, 717-735, (2007) · Zbl 1173.35523
[7] Rossi, J.D.; Souplet, P., Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system, Differential integral equations, 18, 405-418, (2005) · Zbl 1212.35219
[8] Souplet, Ph.; Tayachi, S., Optimal condition for non-simultaneous blow-up in a reaction – diffusion system, J. math. soc. Japan, 56, 571-584, (2004) · Zbl 1059.35049
[9] Wang, M.X., Blowup estimates for a semilinear reaction diffusion system, J. math. anal. appl., 257, 46-51, (2001) · Zbl 0990.35065
[10] Zheng, S.N., Nonexistence of positive solutions to a semilinear elliptic system and blow-up estimates for a reaction – diffusion system, J. math. anal. appl., 232, 293-311, (1999) · Zbl 0935.35042
[11] Lin, Z.G.; Xie, C.H.; Wang, M.X., The blow-up rate of positive solutions of a parabolic system, Northeast. math. J., 13, 327-378, (1997)
[12] Wang, M.X., Blow-up rate estimates for semilinear parabolic systems, J. differential equations, 170, 317-324, (2001) · Zbl 0979.35065
[13] Wang, M.X., Global existence and finite time blow up for a reaction – diffusion system, Z. angew. math. phys., 51, 160-167, (2000) · Zbl 0984.35088
[14] Li, H.L.; Wang, M.X., Properties of blow-up solutions to a parabolic system with nonlinear localized terms, Discrete contin. dyn. syst., 13, 683-700, (2005) · Zbl 1077.35056
[15] Li, F.C.; Huang, S.X.; Xie, C.H., Global existence and blow-up of solutions to a nonlocal reaction – diffusion system, Discrete contin. dyn. syst., 9, 1519-1532, (2003) · Zbl 1043.35069
[16] Cannon, J.R.; Yin, H.M., A class of non-linear non-classical parabolic equations, J. differential equations, 79, 266-288, (1989) · Zbl 0702.35120
[17] Pao, C.V., Blowing-up of solution for a nonlocal reaction – diffusion problem in combustion theory, J. math. anal. appl., 166, (1992), 591C600 · Zbl 0762.35049
[18] Souplet, Ph., Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. differential equations, 153, 374-406, (1999) · Zbl 0923.35077
[19] Zheng, S.N.; Kong, L.H., Roles of weight functions in a nonlinear nonlocal parabolic system, Nonlinear anal., 68, 2406-2416, (2008) · Zbl 1138.35340
[20] Friedman, A., Partial differential equations of parabolic type, (1964), Prentice-Hall, Inc Englewood Cliffs, NJ · Zbl 0144.34903
[21] Lieberman, G.M., Second order parabolic differential equations, (1996), World Scientific Publishing Co., Inc River Edge, NJ · Zbl 0884.35001
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