Skip to content

Advertisement

Boundary Value Problems

Boundary Value Problems Cover Image
Research Article
Open Access

Properties of Positive Solution for Nonlocal Reaction-Diffusion Equation with Nonlocal Boundary

Boundary Value Problems20072007:064579

https://doi.org/10.1155/2007/64579

©  Yulan Wang et al. 2007

Received: 21 January 2007

Accepted: 11 April 2007

Published: 30 May 2007

Abstract

This paper considers the properties of positive solutions for a nonlocal equation with nonlocal boundary condition on . The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we establish the uniform blow-up estimates for the blow-up solution.

Keywords

Boundary ConditionDifferential EquationPartial Differential EquationOrdinary Differential EquationFunctional Equation

[123456789101112131415161718192021]

Authors’ Affiliations

(1)
School of Mathematics and Computer Engineering, Xihua University, Chengdu, China
(2)
Department of Mathematics, Sichuan University, Chengdu, China
(3)
School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, China

References

  1. Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, NY, USA; 1992:xvi+777.MATHGoogle Scholar
  2. Souplet P: Blow-up in nonlocal reaction-diffusion equations. SIAM Journal on Mathematical Analysis 1998,29(6):1301–1334. 10.1137/S0036141097318900MATHMathSciNetView ArticleGoogle Scholar
  3. Souplet P: Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. Journal of Differential Equations 1999,153(2):374–406. 10.1006/jdeq.1998.3535MATHMathSciNetView ArticleGoogle Scholar
  4. Wang M, Wang Y: Properties of positive solutions for non-local reaction-diffusion problems. Mathematical Methods in the Applied Sciences 1996,19(14):1141–1156. 10.1002/(SICI)1099-1476(19960925)19:14<1141::AID-MMA811>3.0.CO;2-9MATHMathSciNetView ArticleGoogle Scholar
  5. Souplet P: Uniform blow-up profile and boundary behaviour for a non-local reaction-diffusion equation with critical damping. Mathematical Methods in the Applied Sciences 2004,27(15):1819–1829. 10.1002/mma.567MATHMathSciNetView ArticleGoogle Scholar
  6. Day WA: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quarterly of Applied Mathematics 1983,40(4):468–475.MATHGoogle Scholar
  7. Day WA: Heat Conduction within Linear Thermoelasticity, Springer Tracts in Natural Philosophy. Volume 30. Springer, New York, NY, USA; 1985:viii+83.View ArticleGoogle Scholar
  8. Friedman A: Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Quarterly of Applied Mathematics 1986,44(3):401–407.MATHMathSciNetGoogle Scholar
  9. Deng K: Comparison principle for some nonlocal problems. Quarterly of Applied Mathematics 1992,50(3):517–522.MATHMathSciNetGoogle Scholar
  10. Seo S: Blowup of solutions to heat equations with nonlocal boundary conditions. Kobe Journal of Mathematics 1996,13(2):123–132.MATHMathSciNetGoogle Scholar
  11. Seo S: Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions. Pacific Journal of Mathematics 2000,193(1):219–226. 10.2140/pjm.2000.193.219MATHMathSciNetView ArticleGoogle Scholar
  12. Lin Z, Liu Y: Uniform blowup profiles for diffusion equations with nonlocal source and nonlocal boundary. Acta Mathematica Scientia. Series B 2004,24(3):443–450.MATHMathSciNetGoogle Scholar
  13. Carlson DE: Linear thermoelasticity. In Encyclopedia of Physics. Volume vIa/2. Springer, Berlin, Germany; 1972:319.Google Scholar
  14. Pao CV: Dynamics of reaction-diffusion equations with nonlocal boundary conditions. Quarterly of Applied Mathematics 1995,53(1):173–186.MATHMathSciNetGoogle Scholar
  15. Pao CV: Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. Journal of Computational and Applied Mathematics 1998,88(1):225–238. 10.1016/S0377-0427(97)00215-XMATHMathSciNetView ArticleGoogle Scholar
  16. Pao CV: Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. Journal of Computational and Applied Mathematics 2001,136(1–2):227–243. 10.1016/S0377-0427(00)00614-2MATHMathSciNetView ArticleGoogle Scholar
  17. Wang Y, Mu C, Xiang Z: Blowup of solutions to a porous medium equation with nonlocal boundary condition. to appear in Applied Mathematics and ComputationGoogle Scholar
  18. Xiang Z, Hu X, Mu C: Neumann problem for reaction-diffusion systems with nonlocal nonlinear sources. Nonlinear Analysis 2005,61(7):1209–1224. 10.1016/j.na.2005.01.098MATHMathSciNetView ArticleGoogle Scholar
  19. Xiang Z, Chen Q, Mu C: Blowup properties for several diffusion systems with localised sources. The ANZIAM Journal 2006,48(1):37–56. 10.1017/S1446181100003400MATHMathSciNetView ArticleGoogle Scholar
  20. Yin H-M: On a class of parabolic equations with nonlocal boundary conditions. Journal of Mathematical Analysis and Applications 2004,294(2):712–728. 10.1016/j.jmaa.2004.03.021MATHMathSciNetView ArticleGoogle Scholar
  21. Zhou J, Mu C, Li Z: Blowup for degenerate and singular parabolic system with nonlocal source. Boundary Value Problems 2006, 2006: 19 pages.MathSciNetView ArticleGoogle Scholar

Copyright

© Yulan Wang et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement