DOI QR코드

DOI QR Code

GLOBAL EXISTENCE AND BLOW-UP FOR A DEGENERATE REACTION-DIFFUSION SYSTEM WITH NONLINEAR LOCALIZED SOURCES AND NONLOCAL BOUNDARY CONDITIONS

  • LIANG, FEI (DEPARTMENT OF MATHEMATICS XI AN UNIVERSITY OF SCIENCE AND TECHNOLOGY)
  • 투고 : 2014.02.26
  • 발행 : 2016.01.01

초록

This paper deals with a degenerate parabolic system with coupled nonlinear localized sources subject to weighted nonlocal Dirichlet boundary conditions. We obtain the conditions for global and blow-up solutions. It is interesting to observe that the weight functions for the nonlocal Dirichlet boundary conditions play substantial roles in determining not only whether the solutions are global or blow-up, but also whether the blowing up occurs for any positive initial data or just for large ones. Moreover, we establish the precise blow-up rate.

키워드

과제정보

연구 과제 주관 기관 : China NSF, China Postdoctoral Science Foundation, Shanxi Provincial Postdoctoral Science Foundation, Xi An University of Science and Technology

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. R. Anderson and K. Deng, Global existence for degenerate parabolic equations with a nonlocal 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
  3. D. G. Aronson, M. G. Crandall, and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), no. 10, 1001-1022. https://doi.org/10.1016/0362-546X(82)90072-4
  4. D. E. Carlson, Linear Thermoelasticity, Encyclopedia, Vol. Via/2, Springer Berlin 1972.
  5. Y. P. Chen and C. H. Xie, Blow-up for a porous medium equation with a localized source, Appl. Math. Comput. 159 (2004), no. 1, 79-93. https://doi.org/10.1016/j.amc.2003.10.032
  6. W. A. Day, Extensions of property of heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982), 319-330. https://doi.org/10.1090/qam/678203
  7. W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 40 (1983), no. 4, 468-475. https://doi.org/10.1090/qam/693879
  8. K. Deng and H. A. Levien, 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
  9. W. B. Deng, Global existence and finite time blow up for a degenerate reaction-diffusion system, Nonlinear Anal. 60 (2005), no. 5, 977-991. https://doi.org/10.1016/j.na.2004.10.016
  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. J. I. Diaz and R. Kerser, On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium, J. Differential Equations 69 (1987), no. 3, 368-403. https://doi.org/10.1016/0022-0396(87)90125-2
  12. E. Dibenedetto, Degenerate Parabolic Equations, Springer, New York, 1993.
  13. Lili. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006), no. 1, 304-320. https://doi.org/10.1016/j.jmaa.2005.11.052
  14. Z. W. Duan, W. B. Deng, and C. H. Xie, Uniform blow-up profile for a degenerate parabolic system with nonlocal source, Comput. Math. Appl. 47 (2004), no. 6-7, 977- 995. https://doi.org/10.1016/S0898-1221(04)90081-8
  15. 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
  16. A. Friedman and B. Mcleod, Blow-up of positive solutions of similinear heat equations, Indiana Univ. Math. J. 34 (1985), no. 2, 425-447. https://doi.org/10.1512/iumj.1985.34.34025
  17. H. A. Levien, The role of critical exponents in blow-up theorems, SIAM Rev 32 (1990), 262-288. https://doi.org/10.1137/1032046
  18. H. A. Levien and P. E. Sacks, Some existence and nonexistence theorems for solutions of degenerate parabolic equations, J. Differential Equations 52 (1984), 135-161. https://doi.org/10.1016/0022-0396(84)90174-8
  19. F. C. Li and C. H. Xie, Global existence and blow-up for a nonlinear porous medium equation, Appl. Math. Lett. 16 (2003), 185-192. https://doi.org/10.1016/S0893-9659(03)80030-7
  20. F. C. Li, Existence and blow-up for a degenerate parabolic equation with nonlocal source, Nonlinear Anal. 52 (2003), 523-534. https://doi.org/10.1016/S0362-546X(02)00119-0
  21. H. L. Li and M. X. Wang, Blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions, J. Math. Anal. Appl. 304 (2005), no. 1, 96-114. https://doi.org/10.1016/j.jmaa.2004.09.020
  22. 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
  23. 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
  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. M. X. Wang, Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions, Appl. Math. Lett. 16 (2003), no. 2, 169-175.
  26. L. Z. Zhao and S. N. Zheng, Critical exponent and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources, J. Math. Anal. Appl. 292 (2004), no. 2, 621-635. https://doi.org/10.1016/j.jmaa.2003.12.011
  27. 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