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Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

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Abstract

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.

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References

  1. de Jager E M, Jiang F R. The theory of singular perturbation[M]. Amsterdam: North-Holland Publishing Co., 1996.

    Google Scholar 

  2. Ni Weiming, Wei Juncheng. On positive solution concentrating on spheres for the Gierer-Meinhardt system[J]. J Diff Eqns, 2006, 221(1):158–189.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  9. Mo Jiaqi, Lin Wantao. A nonlinear singular perturbed problem for reaction diffusion equations with boundary perturbation[J]. Acta Math Appl Sinica, 2005, 21(1):101–104.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  11. Mo Jiaqi, Zhu Jiang, Wang Hui. Asymptotic behavior of the shock solution for a class of nonlinear equations[J]. Progress in Natural Sci, 2003, 13(9):768–770.

    Article  MATH  MathSciNet  Google Scholar 

  12. Mo Jiaqi, Lin Wantao, Zhu Jiang. A variational iteration solving method for ENSO mechanism[J]. Progress in Natural Sci, 2004, 14(12):1126–1128.

    Article  MATH  MathSciNet  Google Scholar 

  13. Mo Jiaqi, Wang Hui, Lin Wantao. Varitional iteration solving method for El Nino phenomenon atmpspheric physics of nonlinear model[J]. Acta Oceanol Sinica, 2005, 24(5): 35–38.

    Google Scholar 

  14. Mo Jiaqi, Wang Hui, Lin Wantao. Singularly perturbed solution of a sea-air oscillator model for the ENSO[J]. Chin Phys, 2006, 15(7):1450–1453.

    Article  Google Scholar 

  15. Mo Jiaqi, Wang Hui, Lin Wantao, Lin Yihua. Varitional iteration method foe solving the mechanism of the equatorial Eastern Pacific El Nino-Southern Oscillation[J]. Chin Phys, 2006, 15(4):671–675.

    Article  MathSciNet  Google Scholar 

  16. Mo Jiaqi, Wang Hui, Lin Wantao. A delayed sea-air oscillator coupling model for the ENSO[J]. Acta Phys Sinica, 2006, 55(7):3229–3232 (in Chinese).

    MathSciNet  Google Scholar 

  17. Mo Jiaqi, Wang Hui. A class of nonlinear nomlocal singularly perturbed problems for reaction diffusion equations[J]. J Biomathematics, 2002, 17(2):143–148.

    MathSciNet  Google Scholar 

  18. Mo Jiaqi, Bionomics dynamic model of human groups fot HIV transmission[J]. Acta Ecologica Sinica, 2006, 26(1):104–107 (in Chinese).

    Google Scholar 

  19. Protter M H, Weinberger H F. Maximum principles in differential equations[M]. New York: Prentice-Hall Inc, 1967.

    Google Scholar 

  20. Pao C V. Comparison methods and stability analysis of reaction diffusion systems[J]. Lecture Notes in Pure and Appl Math, 1994, 162(1):277–292.

    MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Jia-qi Mo  (莫嘉琪)

  2. Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, Zhejiang Province, P. R. China

    Jia-qi Mo  (莫嘉琪)

  3. Division of Computational Science, E-Institutes of Shanghai Universities at Shanghai Jiaotong University, Shanghai, 200240, P. R. China

    Jia-qi Mo  (莫嘉琪)

Authors
  1. Jia-qi Mo  (莫嘉琪)
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Correspondence to Jia-qi Mo  (莫嘉琪).

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Communicated by ZHANG Hong-qing

Project supported by the National Natural Science Foundation of China (Nos. 40676016, 10471039), the National Key Basic Research Special Foundation of China (No. 2004CB418304), the Key Basic Research Foundation of the Chinese Academy of Sciences (No. KZCX3-SW-221) and in part by EInstitutes of Shanghai Municipal Education Commission (No. E03004)

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Mo, Jq. Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation. Appl. Math. Mech.-Engl. Ed. 29, 1105–1110 (2008). https://doi.org/10.1007/s10483-008-0814-x

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  • DOI: https://doi.org/10.1007/s10483-008-0814-x

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2000 Mathematics Subject Classification

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