Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Global Attractivity and Oscillations in a Nonlinear Impulsive Parabolic Equation with Delay

  • Wang, Xiao (Department of Mathematics and System Science, College of Science, National University of Defense Technology) ;
  • Li, Zhixiang (Department of Mathematics and System Science, College of Science, National University of Defense Technology)
  • Received : 2006.07.25
  • Published : 2008.12.31
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Abstract

Global attractivity and oscillatory behavior of the following nonlinear impulsive parabolic differential equation which is a general form of many population models $$\array{\{{{\frac {{\partial}u(t,x)}{{\partial}t}=\Delta}u(t,x)-{\delta}u(t,x)+f(u(t-\tau,x)),\;t{\neq}t_k,\\u(t^+_k,x)-u(t_k,x)=g_k(u(t_k,x)),\;k{\in}I_\infty,}\;\;\;\;\;\;\;\;(*)$$ are considered. Some new sufficient conditions for global attractivity and oscillation of the solutions of (*) with Neumann boundary condition are established. These results no only are true but also improve and complement existing results for (*) without diffusion or impulses. Moreover, when these results are applied to the Nicholson's blowflies model and the model of Hematopoiesis, some new results are obtained.

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References

  1. D. D. Bainov and E. Minchev, Oscillation of the solutions of impulsive parabolic equations, J. Comput. Appl. Math., 69(1996), 207-214. https://doi.org/10.1016/0377-0427(95)00040-2
  2. C. Cui, M. Zou, A. Liu and L. Xiao, Oscillation of nonlinear impulsive parbolic differential equations with several delays, Ann. of Diff. Eqs., 21(1)(2005), 1-7.
  3. T. Ding, Asymptotic behavior of solution of some retarded differential equations, Scientia Sinica(A), 25(4) (1982), 363-370.
  4. L. H. Erbe, H. I. Freedman, X. Liu and J. H.Wu, Comparison principles for impulsive parabolic equations with applications to models of single species growth, J. Austral. Math. Soc. Ser. B32(1991), 382-400.
  5. X. Fu, L. Shiau, Oscillation criteria for impulsive parabolic boundary value problem with delay,Applied Mathematics and Computation, 153(2004), 587-599. https://doi.org/10.1016/S0096-3003(03)00659-3
  6. X. Fu, X. Liu and S. Sivaloganathan, Oscillation criteria for impulsive parabolic differential equations with delay, J. Math. Anal. Appl., 268(2002), 647-664. https://doi.org/10.1006/jmaa.2001.7840
  7. X. Fu, B. Q. Yan and Y. S. Liu, Introduction to impulsive differential systems(in Chinese), Science Press, Beijing, 2005.
  8. W. Gao and J. Wang, Estimates of solutions of impulsive parabolic equations under Neumann boundary condition, J. Math. Anal. Appl., 283(2003), 478-490. https://doi.org/10.1016/S0022-247X(03)00275-0
  9. M. Kirane and Y. V. Rogovchenko, Comparison results for systems of impulse parabolic equations with applications to population dynamics, Nonlinear Anal. TMA., 28(2)(1997), 263-276. https://doi.org/10.1016/0362-546X(95)00159-S
  10. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
  11. M. R. S. Kulenovic, G. Ladas and A. Meimaridou, On oscillation of nonlinear delay differential equations. Quarterly Of Applied Mathematics, XLV(1)(1987), 155-164.
  12. W. Li, On the forced oscillation of solutions for systems of impulsive parabolic differential equations with several delays, J. Comput. Appl. Math., 181(2005), 46-57. https://doi.org/10.1016/j.cam.2004.11.016
  13. W. Li ,M. Han and F. Meng, Necessary and sufficient conditions for oscillation of impulsive parabolic differential equations with delays, Applied Mathematics Letters, 18(2005), 1149-1155. https://doi.org/10.1016/j.aml.2005.01.001
  14. M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197(1977), 287-289. https://doi.org/10.1126/science.267326
  15. R. Redlinger. Exitence theorems for Semiliner parabolic Systems with Functional, Nonlinear Anal. TMA., 8(1984), 667-682. https://doi.org/10.1016/0362-546X(84)90011-7
  16. R. Redlinger, On Volterra's population equation with diffusion, SIAM J. Math. Anal., 16(1985), 135-142. https://doi.org/10.1137/0516008
  17. S. H. Saker, Oscillation and global attractivity in hematopoiesis model with delay time, Applied Mathematics and Computation, 136(2003), 241-250. https://doi.org/10.1016/S0096-3003(02)00035-8
  18. O. O. Struk and V. I. Tkachenko, On impulsive lotka-volterra systems with diffusion, Ukrainian Mathematical Journal, 54(4)(2002), 629-646. https://doi.org/10.1023/A:1021039528818
  19. J. R. Yan and Ch. Kou, Oscillation of Solutions of Impulsive Delay Differential Equations, J. Math. Anal. Appl., 254(2001), 358-370. https://doi.org/10.1006/jmaa.2000.7112
  20. Y. Yang, J. W. -H. So, Dynamics for the diffusive Nicholson blowflies equation, in: Proceedings of the International Conference on Dynamical Systems and Differential Equations, held in Springfield, Missouri, USA, May 29-June 1, 1996.

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