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Asymptotic Solution of Activator Inhibitor Systems for Nonlinear Reaction Diffusion Equations

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Abstract

A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.

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References

  1. E. M. de Jager and F. R. Jiang, The Theory of Singular Perturbation, North-Holland Publishing Co., Amsterdam, 1996.

    Google Scholar 

  2. W. M. Ni and J. C. Wei, On positive solution concentrating on spheres for the Gierer-Meinhardt system, J. Diff. Eqns., 2006, 221(1): 158–189.

    Article  Google Scholar 

  3. F. Zhang, Coexistence of a pulse and multiple spikes and transition layers in the standing waves of a reaction-diffusion system, J. Diff. Eqns., 2004, 205(1): 77–155.

    Article  Google Scholar 

  4. R. Z. Khasminskii and G. Yin, Limit behavior of two-time-scale diffusion revisited, J. Diff. Eqns., 2005, 212(1): 85–113.

    Article  Google Scholar 

  5. I. Marques, Existence and asymptotic behavior of solutions for a class of nonlinear elliptic equations with Neumann condition, Nonlinear Anal., 2005, 61(1): 21–40.

    Article  Google Scholar 

  6. A. S. Bobkova, The behavior of solutions of multidimensional singularly perturbed system with one fast variable, Diff. Eqns., 2005, 41(1): 23–32.

    Google Scholar 

  7. J. Q. Mo, A singularly perturbed nonlinear boundary value problem, J. Math. Anal. Appl., 1993, 178(1): 289–293.

    Article  Google Scholar 

  8. J. Q. Mo, Singular perturbation for a class of nonlinear reaction diffusion systems, Science in China Ser. A, 1989, 32(11): 1306–1315.

    Google Scholar 

  9. J. Q. Mo and W. T. Lin, A nonlinear singular perturbed problem for reaction diffusion equations with boundary perturbation, Acta Math. Appl. Sinica, 2005, 21(1): 101–104.

    Article  Google Scholar 

  10. J. Q. Mo and S. Shao, The singularly perturbed boundary value problems for higher-order semilinear elliptic equations, Advances in Math., 2001, 30(2): 141–148.

    Google Scholar 

  11. J. Q. Mo, J. Zhu, and H. Wang, Asymptotic behavior of the shock solution for a class of nonlinear equations, Progress in Natural Sci., 2003, 13(9): 768–770.

    Article  Google Scholar 

  12. J. Q. Mo and X. L. Han, Asymptotic shock solution for a nonlinear equation, Acta Math. Sci., 2004, 24(2): 164–167.

    Google Scholar 

  13. J. Q. Mo, W. T. Lin, and J. Zhu, A variational iteration solving method for ENSO mechanism, Progress in Natural Sci., 2004, 14(12): 1126–1128.

    Article  Google Scholar 

  14. J. Q. Mo, H. Wang, W. T. Lin, and Y. H. Lin, Sea-air oscillator model for equatorial eastern Pacific SST, Acta Phys. Sinica, 2006, 55(1): 6–9.

    Google Scholar 

  15. J. Q. Mo, H. Wang, W. T. Lin, and Y. H. Lin, Varitional iteration method thr mechanism of the equatorial eastern Pacific El Nino-Southern Oscillation, Chin. Phys., 2006, 15(4): 671–675.

    Article  Google Scholar 

  16. M. Taniguchi, Instability of planar travling waves in bistable reaction-diffusion systems, Discrete and Continuous Dynamicl Systems, 2003, 3B(1): 21–44.

    Google Scholar 

  17. M. A. Collins and J. Ross, Chemical relaxation pulses and waves, analysis of lowest order multiple time scales expansion, J. Chem. Phys., 1978, 68(10): 3774–3784.

    Article  Google Scholar 

  18. P. Ortoleva and J. Ross, Theory of propagation of discontinuities in kinetic systems with multiple time scales: Fronts, front multiplicity, and pulses, J. Chem. Phys., 1975, 63(9): 3398–3408.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Anhui Normal University, Wuhu, 241000, China

    Jiaqi Mo

  2. Division of Computational Science, E-Institutes of Shanghai Universities at SJTU, Shanghai, 200240, China

    Jiaqi Mo

  3. LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, 100029, China

    Wantao Lin

Authors
  1. Jiaqi Mo
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  2. Wantao Lin
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Correspondence to Jiaqi Mo.

Additional information

This research is supported by the National Natural Science Foundation of China under Grant Nos. 40676016 and 10471039, the National Key Project for Basics Research under Grant Nos. 2003CB415101-03 and 2004CB418304, the Key Project of the Chinese Academy of Sciences under Grant No. KZCX3-SW-221, and in part by E-Insitutes of Shanghai Municipal Education Commission under Grant No. E03004.

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Mo, J., Lin, W. Asymptotic Solution of Activator Inhibitor Systems for Nonlinear Reaction Diffusion Equations. J. Syst. Sci. Complex. 21, 119–128 (2008). https://doi.org/10.1007/s11424-008-9071-4

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  • DOI: https://doi.org/10.1007/s11424-008-9071-4

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