Frontiers of Mathematics in China

, Volume 7, Issue 1, pp 97–116 | Cite as

A degenerate parabolic system with localized sources and nonlocal boundary condition

Research Article
First Online:
Received:
Accepted:

Abstract

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.

Keywords

Nonlocal boundary condition localized sources blow-up rate Porous medium equation 

MSC

35B40 35K50 35K57 35K60 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson J R. Local existence and uniqueness of solutions of degenerate parabolic equations. Comm Partial Differential Equations, 1991, 16: 105–143CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Andreucci D, Herrero M A, Velaázquez J L. Liouville theorems and blow up behavior in semilinear reaction-diffusion systems. Ann Inst H Poincaré Anal Non Linéaire, 1997, 14: 1–53CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Aronson D G, Crandall M G, Peletier L A. Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal, 1982, 6: 1001–1022CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bebernes J, Bressan A, Lacey A. Total explosions versus single-point blow-up. J Differential Equations, 1988, 73: 30–44CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cantrell R S, Cosner C. Diffusive logistic equation with indefinite weights: Population models in disrupted environments II. SIAM J Math Anal, 1991, 22: 1043–1064CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chadam J M, Peirce A, Yin H M. The blowup property of solutions to some diffusion equations with localized nonlinear reactions. J Math Anal Appl, 1992, 169: 313–328CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chen Y, Xie C. Blow-up for a porous medium equation with a localized source. Appl Math Comput, 2004, 159: 79–93CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Chen Y P, Liu Q L, Gao H J. Uniform blow-up rate for diffusion equations with localized nonlinear source. J Math Anal Appl, 2006, 320: 771–778CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chen Y P, Liu Q L, Gao H J. Boundedness of global solutions of a porous medium equation with a localized source. Nonlinear Anal, 2006, 64: 2168–2182CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Chen Y P, Liu Q L, Gao H J. Boundedness of global positive solutions of a porous medium equation with a moving localized source. J Math Anal Appl, 2007, 333: 1008–1023CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Cui Z J, Yang Z D. Boundedness of global solutions for a nonlinear degenerate parabolic (porous medium) system, with localized sources. Appl Math Comput, 2008, 198: 882–895CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Cui Z J, Yang Z D. Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition. J Math Anal Appl, 2008, 342: 559–570CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Day W A. Extensions of property of heat equation to linear thermoelasticity and other theories. Quart Appl Math, 1982, 40: 319–330MATHMathSciNetGoogle Scholar
  14. 14.
    Day W A. A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quart Appl Math, 1983, 41: 468–475Google Scholar
  15. 15.
    Day W A. Heat Conduction within Linear Thermoelasticity. Springer Tracts in Natural Philosophy, Vol 30. New York: Springer, 1985CrossRefMATHGoogle Scholar
  16. 16.
    Deng K. Comparison principle for some nonlocal problems. Quart Appl Math, 1992, 50: 517–522MATHMathSciNetGoogle Scholar
  17. 17.
    Deng W B. Global existence and finite time blow up for a degenerate reaction-diffusion system. Nonlinear Anal, 2005, 60: 977–991CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Diaz J I. On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium. J Differential Equations, 1987, 69: 368–403CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Duan Z W, Deng W B, Xie C H. Uniform blow-up profile for a degenerate parabolic system with nonlocal source. Comput Math Appl, 2004, 47: 977–995CrossRefMathSciNetGoogle Scholar
  20. 20.
    Friedman A, Mcleod B. Blow-up of positive solutions of semilinear heat equations. Indiana Univ Math J, 1985, 34: 425–447CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Friedman A. Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Quart Appl Math, 1986, 44: 401–407MATHMathSciNetGoogle Scholar
  22. 22.
    Furter J, Grinfeld M. Local vs. nonlocal interactions in population dynamics. J Math Biol, 1989, 27: 65–80CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Han Y Z, Gao W J. Global existence and blow-up for a class of degenerate parabolic systems with localized source. Acta Appl Math, DOI: 10.1007/s10440-010-9563-9Google Scholar
  24. 24.
    Kong L H, Wang M X. Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries. Science in China, Ser A, 2007, 50: 1251–1266CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Li F J, Liu B C. Non-simultaneous blow-up in parabolic equations coupled via localized sources. Appl Math Lett (to appear)Google Scholar
  26. 26.
    Li H L, Wang M X. Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms. Sci China, Ser A Math, 2005, 48: 185–197CrossRefMATHGoogle Scholar
  27. 27.
    Li H L, Wang M X. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete Contin Dyn Syst, 2005, 13: 683–700CrossRefMathSciNetGoogle Scholar
  28. 28.
    Li J, Cui Z J, Mu C J. Global existence and blow-up for degenerate and singular parabolic system with localized sources. Appl Math Comput, 2008, 199: 292–300CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Li Y X, Gao W J, Han Y Z. Boundedness of global solutions for a porous medium system with moving localized sources. Nonlinear Anal, 2010, 72: 3080–3090CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Lin Z G, Liu Y R. Uniform blow-up profiles for diffusion equations with nonlocal source and nonlocal boundary. Acta Math Sci, Ser B, 2004, 24: 443–450MATHMathSciNetGoogle Scholar
  31. 31.
    Mi Y S, Mu C L. Blowup properties for nonlinear degenerate diffusion equations with localized sources. PreprintGoogle Scholar
  32. 32.
    Ortoleva P, Ross J. Local structures in chemical reaction with heterogeneous catalysis. J Chem Phys, 1972, 56: 43–97Google Scholar
  33. 33.
    Pao C V. Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press, 1992MATHGoogle Scholar
  34. 34.
    Pao C V. Blowing-up of solution of a nonlocal reaction-diffusion problem in combustion theory. J Math Anal Appl, 1992, 166: 591–600CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Pao C V. Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. Comput Math Appl, 1998, 88: 225–238CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Pedersen M, Lin Z. The profile near blowup time for solutions of diffusion systems coupled with localized nonlinear reactions. Nonlinear Anal, 2002, 50: 1013–1024CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Seo S. Blowup of solutions to heat equations with nonlocal boundary conditions. Kobe Journal of Mathematics, 1996, 13: 123–132MATHMathSciNetGoogle Scholar
  38. 38.
    Souplet P. Blow-up in nonlocal reaction-diffusion equations. SIAM J Math Anal, 1998, 29: 1301–1334CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Souplet P. Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. J Differential Equations, 1999, 153: 374–406CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Wang L, Chen Q. The asymptotic behavior of blow-up solution of localized nonlinear equation. J Math Anal Appl, 1996, 200: 315–321CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Wang M X, Wen Y F, Blow-up properties for a degenerate parabolic system with nonlinear localized sources. J Math Anal Appl, 2008, 343: 621–635CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Wang Y L, Mu C L, Xiang Z Y. Blowup of solutions to a porous medium equation with nonlocal boundary condition. Appl Math Comput, 2007, 192: 579–585CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Wang Y L, Mu C L, Xiang Z Y. Properties of positive solution for nonlocal reactiondiffusion equation with nonlocal boundary. Boundary Value Problems, 2007, Article ID 64579, 12 pagesGoogle Scholar
  44. 44.
    Wang Y L, Xiang Z Y. Blowup analysis for a semilinear parabolic system with nonlocal boundary condition. Boundary Value Problems, 2009, Article ID 516390, 14 pages, doi: 10.1155/2009/516390Google Scholar
  45. 45.
    Yin H M. On a class of parabolic equations with nonlocal boundary conditions. J Math Anal Appl, 2004, 294: 712–728CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Yin Y F. On nonlinear parabolic equations with nonlocal boundary condition. J Math Anal Appl, 1994, 185: 161–174CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Zheng S N, Kong L. Roles of weight functions in a nonlinear nonlocal parabolic system. Nonlinear Anal, 2008, 68: 2406–2416CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Zhao L Z, Zheng S N. Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources. J Math Anal Appl, 2004, 292: 621–635CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Zhou J, Mu C L. Uniform blow-up profiles and boundary layer for a parabolic system with localized sources. Nonlinear Anal, 2008, 69: 24–34CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingChina
  2. 2.College of Mathematics and Computer SciencesYangtze Normal UniversityFulingChina

Personalised recommendations