대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제27권3호
  • /
  • Pages.629-644
  • /
  • 2012
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

A PARABOLIC SYSTEM WITH NONLOCAL BOUNDARY CONDITIONS AND NONLOCAL SOURCES

  • Gao, Wenjie (State Key Laboratory of Automotive Dynamic Simulation Jilin University, Institute of Mathematics Jilin University) ;
  • Han, Yuzhu (Institute of Mathematics Jilin University)
  • 투고 : 2011.01.11
  • 발행 : 2012.07.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this work, the authors study the blow-up properties of solutions to a parabolic system with nonlocal boundary conditions and nonlocal sources. Conditions for the existence of global or blow-up solutions are given. Global blow-up property and precise blow-up rate estimates are also obtained.

키워드

참고문헌

  1. J. R. Anderson, Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), no. 1, 105-143. https://doi.org/10.1080/03605309108820753
  2. J. Bebernes, A. Bressan, and A. Lacey, Total blow-up versus single point blow-up, J. Differential Equations 73 (1988), no. 1, 30-44. https://doi.org/10.1016/0022-0396(88)90116-7
  3. S. Carl and V. Lakshmikantham, Generalized quasilinearization method for reaction- diffusion equations under nonlinear and nonlocal flux conditions, J. Math. Anal. Appl. 271 (2002), no. 1, 182-205. https://doi.org/10.1016/S0022-247X(02)00114-2
  4. D. E. Carlson, Linear thermoelasticity, in Encyclopedia of Physics, Springer, Berlin, 1972.
  5. Y. J. Chen and M. X. Wang, A class of nonlocal and degenerate quasilinear parabolic system not in divergence form, Nonlinear Anal. 71 (2009), no. 7-8, 3530-3537. https://doi.org/10.1016/j.na.2009.02.016
  6. Z. J. Cui and Z. D. Yang, Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition, J. Math. Anal. Appl. 342 (2008), no. 1, 559-570. https://doi.org/10.1016/j.jmaa.2007.11.055
  7. W. A. Day, Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982/83), no. 3, 319-330. https://doi.org/10.1090/qam/678203
  8. W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 40 (1982/83), no. 4, 468-475.
  9. K. Deng, Comparison principle for some nonlocal problems, Quart. Appl. Math. 50 (1992), no. 3, 517-522. https://doi.org/10.1090/qam/1178431
  10. W. B. Deng, Y. X. Li, and C. H. Xie, Blow-up and global existence for a nonlocal degenerate parabolic system, J. Math. Anal. Appl. 277 (2003), no. 1, 199-217. https://doi.org/10.1016/S0022-247X(02)00533-4
  11. L. L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math. 202 (2007), no. 2, 237-247. https://doi.org/10.1016/j.cam.2006.02.028
  12. A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44 (1986), no. 3, 401-407. https://doi.org/10.1090/qam/860893
  13. 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
  14. L. J. Jiang and H. L. Li, Uniform blow-up profiles and boundary layer for a parabolic system with nonlocal sources, Math. Comput. Modelling 45 (2007), no. 7-8, 814-824. https://doi.org/10.1016/j.mcm.2006.07.023
  15. Z. G. Lin and Y. R. Lin, Uniform blowup profile for diffusion equations with nonlocal source and nonlocal boundary, Acta Math. Sci. Ser. B Engl. Ed. 24 (2004), no. 3, 443-450.
  16. C. V. Pao, Dynamics of reaction-diffusion equations with nonlocal boundary conditions, Quart. Appl. Math. 50 (1995), no. 1, 173-186.
  17. 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
  18. C. V. Pao, Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 136 (2001), no. 1-2, 227-243. https://doi.org/10.1016/S0377-0427(00)00614-2
  19. S. Seo, Blowup of solutions to heat equations with nonlocal boundary conditions, Kobe J. Math. 13 (1996), no. 2, 123-132.
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. 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
  26. 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