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Blowup properties for several diffusion systems with localised sources

Published online by Cambridge University Press:  17 February 2009

Zhaoyin Xiang ,
Qiong Chen  and
Chunlai Mu
Zhaoyin Xiang
Affiliation:
School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, P. R., China; e-mail: zhaoyin-xiang@sohu.com. Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R., China.
Qiong Chen
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R., China.
Chunlai Mu
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R., China.
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Abstract

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This paper investigates the Cauchy problem for two classes of parabolic systems with localised sources. We first give the blowup criterion, and then deal with the possibilities of simultaneous blowup or non-simultaneous blowup under some suitable assumptions. Moreover, when simultaneous blowup occurs, we also establish precise blowup rate estimates. Finally, using similar ideas and methods, we shall consider several nonlocal problems with homogeneous Neumann boundary conditions.

Keywords

diffusion systemlocalised sourceCauchy problemsimultaneous blowupblowup estimates.
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 1 , July 2006 , pp. 37 - 56
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Bimpong-Bota, K., Ortoleva, P. and Ross, J., “Far-from-equilibrum phenomena at local sites of reactions”, J. Chem. Phys. 60 (1974) 3124–3133.CrossRefGoogle Scholar
[2]Cannon, J. R. and Yin, H.-M., “A class of non-linear non-classical parabolic equations”, J. Differential Equations 79 (1989) 266–288.CrossRefGoogle Scholar
[3]Chadam, J. M., Peirce, A. and Yin, H.-M., “The blow-up property of solutions to some diffusion equations with localized nonlinear reactions”, J. Math. Anal. Appl. 169 (1992) 313–328.CrossRefGoogle Scholar
[4]Chadam, J. M. and Yin, H.-M., “An iteration procedure for a class of integrodifferential equations of parabolic type”, J. Integral Equations Appl. 2 (1989) 31–47.CrossRefGoogle Scholar
[5]Chadam, J. M. and Yin, H.-M., “A diffusion equation with localized chemical reactions”, Proc. Edinb. Math. Soc. 37 (1993) 101–118.CrossRefGoogle Scholar
[6]Deng, K. and Levine, H. A., “The role of critical exponents in blowup theorems: The sequel”, J. Math. Anal. Appl. 243 (2000) 85–126.CrossRefGoogle Scholar
[7]Friedman, A., Partial differential equations of parabolic type (Prentice-Hill, Englewood Cliffs, N.J., 1964).Google Scholar
[8]Galaktionov, V. A. and Vázquez, J. L., “The problem of blow-up in nonlinear parabolic equations”, Disc. Cont. Dyn. Systems 8 (2002) 399–433.CrossRefGoogle Scholar
[9]Levine, H. A., “The role of critical exponents in blow-up problems”, SIAM Rev. 32 (1990) 262–288.CrossRefGoogle Scholar
[10]Li, F. C., Huang, S. X. and Xie, C. H., “Global existence and blow-up of solutions to a nonlocal reaction-diffusion system”, Dist. Cont. Dyn. Systems 9 (2003) 1519–1532.Google Scholar
[11]Li, H. L. and Wang, M. X., “Blow-up properties for parabolic systems with localized nonlinear sources”, Appl. Math. Lett. 17 (2004) 771–778.Google Scholar
[12]Li, H. L. and Wang, M. X., “Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms”, Sci. China Ser. A Math. 48 (2005) 185–197.CrossRefGoogle Scholar
[13]Lin, Z., Xie, C. and Wang, M., “The blow-up properties of solutions to a parabolic system with localized nonlinear reactions”, Acta Math. Sci. 18 (1998) 413–420.CrossRefGoogle Scholar
[14]Ortaleva, P. and Ross, J., “Local structures in chemical reactions with heterogeneous catalysis”, J. Chem. Phys. 56 (1972) 4397.CrossRefGoogle Scholar
[15]Pao, C. V., Nonlinear parabolic and elliptic equations (Plenum Press, New York, London, 1992).Google Scholar
[16]Pedersen, M. and Lin, Z., “Coupled diffusion systems with localized nonlinear reactions”, Comput. Math. Appl. 42 (2001) 807–816.CrossRefGoogle Scholar
[17]Souplet, P., “Blow up in nonlocal reaction-diffusion equation”, SIAM J. Math. Anal. 29 (1998) 1301–1334.CrossRefGoogle Scholar
[18]Souplet, P., “Uniform blowup profiles and boundary behavior for diffusion equations with nonlocal nonlinear source”, J. Differential Equations 153 (1999) 374–406.CrossRefGoogle Scholar
[19]Wang, L. and Chen, Q., “The asymptotic behavior of blowup solution of localized nonlinear equation”, J. Math. Anal. Appl. 200 (1996) 315–321.CrossRefGoogle Scholar
[20]Xiang, Z. Y., Hu, X. G. and Mu, C. L., “Neumann problem for reaction-diffusion systems with nonlocal nonlinear sources”, Nonlinear Anal. (TMA) 61 (2005) 1209–1224.CrossRefGoogle Scholar
[21]Zhao, L. Z. and Zheng, S. N., “Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources”, J. Math. Anal. Appl. 292 (2004) 621–635.CrossRefGoogle Scholar