DOI QR코드

DOI QR Code

SOME TYPES OF REACTION-DIFFUSION SYSTEMS WITH NONLOCAL BOUNDARY CONDITIONS

  • Han, Yuzhu (Institute of Mathematics Jilin University) ;
  • Gao, Wenjie (Institute of Mathematics Jilin University)
  • 투고 : 2011.05.10
  • 발행 : 2013.11.30

초록

This paper deals with some types of semilinear parabolic systems with localized or nonlocal sources and nonlocal boundary conditions. The authors first derive some global existence and blow-up criteria. And then, for blow-up solutions, they study the global blow-up property as well as the precise blow-up rate estimates, which has been seldom studied until now.

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참고문헌

  1. J. R. Anderson and K. Deng, Global existence for degenerate parabolic equations with a non-local forcing, Math. Methods Appl. Sci. 20 (1997), no. 13, 1069-1087. https://doi.org/10.1002/(SICI)1099-1476(19970910)20:13<1069::AID-MMA867>3.0.CO;2-Y
  2. J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York, 1989.
  3. K. Bimpong-Bota, P. Ortoleva, and J. Ross, Far-from-equilibrium phenomena at local cites of reaction, J. Chem. Phys. 60 (1974), 3124-3133. https://doi.org/10.1063/1.1681498
  4. D. E. Carlson, Linear thermoelasticity, Encyclopedia, vol. vIa/2, Springer, Berlin, 1972.
  5. J. M. Chadam, A. Peirce, and H. M. Yin, The blowup property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992), no. 2 313-328. https://doi.org/10.1016/0022-247X(92)90081-N
  6. Y. Chen and H. Gao, Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation, J. Math. Anal. Appl. 330 (2007), no. 2, 852-863. https://doi.org/10.1016/j.jmaa.2006.08.014
  7. A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44 (1986), no. 3, 401-407.
  8. A. Friedman and J. B. Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), no. 2, 425-447. https://doi.org/10.1512/iumj.1985.34.34025
  9. W. A. Day, Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982), no. 3, 319-330.
  10. W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 40 (1983), no. 4, 468-475.
  11. K. Deng, Comparison principle for some nonlocal problems, Quart. Appl. Math. 50 (1992), no. 3, 517-522.
  12. K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000), no. 1, 85-126. https://doi.org/10.1006/jmaa.1999.6663
  13. Y. Han and W. Gao, Global existence and blow-up for a class of degenerate parabolic systems with localized source, Acta Appl. Math. 112 (2010), no. 2, 251-261. https://doi.org/10.1007/s10440-010-9563-9
  14. L. H. Kong and M. X. Wang, Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries, Sci. China Ser. A 50 (2007), no. 9, 1251-1266. https://doi.org/10.1007/s11425-007-0105-5
  15. H. A. Levine, The role of critical exponents in blow up theorems, SIAM Rev. 32 (1990), 262-288. https://doi.org/10.1137/1032046
  16. H. L. Li and M. X. Wang, Properties of blow-up solutions to a parabolic system with nonlinear localized terms, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 683-700. https://doi.org/10.3934/dcds.2005.13.683
  17. P. Ortoleva and J. Ross, Local structures in chemical reactions with heterogeneous catal-ysis, J. Chem. Phys. 56 (1972), 4397-4400. https://doi.org/10.1063/1.1677879
  18. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.
  19. C. V. Pao, Dynamics of reaction-diffusion equations with nonlocal boundary conditions, Quart. Appl. Math. 50 (1995), no. 1, 173-186.
  20. C. V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 88 (1998), no. 1, 225-238. https://doi.org/10.1016/S0377-0427(97)00215-X
  21. C. V. Pao, Numerical solutions of reaction-diffusion equations with nonlocal boundary con-ditions, J. Comput. Appl. Math. 136 (2001), no. 1-2, 227-243. https://doi.org/10.1016/S0377-0427(00)00614-2
  22. S. Seo, Blowup of solutions to heat equations with nonlocal boundary conditions, Kobe J. Math. 13 (1996), no. 2, 123-132.
  23. S. Seo, Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions, Pacific J. Math. 193 (2000), no. 1, 219-226. https://doi.org/10.2140/pjm.2000.193.219
  24. P. Souplet, Blow up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29 (1998), no. 6, 1301-1334. https://doi.org/10.1137/S0036141097318900
  25. P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999), no. 2, 374-406. https://doi.org/10.1006/jdeq.1998.3535
  26. Y. L. Wang, C. L. Mu, and Z. Y. Xiang, Blowup of solutions to a porous medium equation with nonlocal boundary condition, Appl. Math. Comput. 192 (2007), no. 2, 579-585. https://doi.org/10.1016/j.amc.2007.03.036
  27. M. X. Wang and Y. M. Wang, Properties of positive solutions for non-local reaction- diffusion problems, Math. Methods Appl. Sci. 19 (1996), no. 14, 1141-1156. https://doi.org/10.1002/(SICI)1099-1476(19960925)19:14<1141::AID-MMA811>3.0.CO;2-9
  28. Y. L. Wang and Z. Y. Xiang, Blowup analysis for a semilinear parabolic system with nonlocal boundary condition, Boundary Value Problems 2009 (2009), Article ID 516390, 14 pages.
  29. Z. Y. Xiang, X. G. Hu, and C. L. Mu, Neumann problem for reaction-diffusion systems with nonlocal nonlinear sources, Nonlinear Anal. 61 (2005), no. 7, 1209-1224. https://doi.org/10.1016/j.na.2005.01.098
  30. H. M. Yin, On a class of parabolic equations with nonlocal boundary conditions, J. Math. Anal. Appl. 294 (2004), no. 2, 712-728. https://doi.org/10.1016/j.jmaa.2004.03.021
  31. Y. F. Yin, On nonlinear parabolic equations with nonloal boundary conditions, J. Math. Anal. Appl. 185 (1994), no. 1, 161-174. https://doi.org/10.1006/jmaa.1994.1239
  32. S. N. Zheng and L. H. Kong, Roles of weight functions in a nonlinear nonlocal parabolic system, Nonlinear Anal. 68 (2008), no. 8, 2406-2416. https://doi.org/10.1016/j.na.2007.01.067