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Roles of weight functions in a nonlinear nonlocal parabolic system. (English) Zbl 1138.35340

Summary: This paper deals with a semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions. We establish the conditions for global and non-global solutions. It is interesting to observe that the weight functions for the nonlocal Dirichlet boundary conditions play substantial roles in determining not only whether the solutions are global or non-global, but also whether (for the non-global solutions) the blowing up occurs for any positive initial data or just for large ones.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35B33 Critical exponents in context of PDEs
35K55 Nonlinear parabolic equations
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[1] 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
[2] Deng, K., Comparison principle for some nonlocal problems, Quart. Appl. Math., 50, 517-522 (1992) · Zbl 0777.35006
[3] Escobedo, M.; Levine, H. A., Critical blow-up and global existence numbers for a weakly coupled systems of reaction-diffusion equations, Arch. Ration. Mech. Anal., 129, 47-100 (1995) · Zbl 0822.35068
[4] Escobedo, M.; Herrero, M. A., A semilinear parabolic system in a bounded domain, Ann. Mat. Pura Appl., CLXV(IV), 315-336 (1993) · Zbl 0806.35088
[5] Friedman, A., Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math., 44, 401-407 (1986) · Zbl 0631.35041
[6] Kaplan, S., On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16, 305-330 (1963) · Zbl 0156.33503
[7] Levine, L. A., A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations, Z. Angew. Math. Phys., 42, 408-430 (1993) · Zbl 0786.35075
[8] 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
[9] Lin, Z. G.; Liu, Y. R., Uniform blow-up profiles for diffusion equations with nonlocal source and nonlocal boundary, Acta Math. Sci., 24B, 443-450 (2004) · Zbl 1065.35150
[10] 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
[11] Pao, C. V., On nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 87, 165-198 (1982) · Zbl 0488.35043
[12] Pao, C. V., Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math., 88, 225-238 (1998) · Zbl 0920.35030
[13] Souplet, P., Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations, 153, 374-406 (1999) · Zbl 0923.35077
[14] Souplet, P., Blow up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29, 1301-1334 (1998) · Zbl 0909.35073
[15] Wang, L. W., The blow-up for weakly coupled reaction-diffusion systems, Proc. Amer. Math. Soc., 129, 89-95 (2000) · Zbl 0958.35064
[16] 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
[17] Wang, M. X.; Wang, Y. M., Properties of positive solutions for non-local reaction-diffusion problems, Math. Methods Appl. Sci., 19, 1141-1156 (1996) · Zbl 0990.35066
[18] Zhao, L. Z.; Zheng, S. N., Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources, J. Math. Anal. Appl., 292, 621-635 (2004) · Zbl 1052.35034
[19] Zheng, S. N., Global existence and global nonexistence of solutions to a reaction-diffusion system, Nonlinear Anal., 39, 327-340 (2000) · Zbl 0955.35039
[20] Zheng, S. N.; Wang, L. D., Blow-up rate and profile for a degenerate parabolic system coupled via nonlocal sources, Comput. Math. Appl., 52, 1387-1402 (2006) · Zbl 1132.35403
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