Open Access

On Comparison Principles for Parabolic Equations with Nonlocal Boundary Conditions

Boundary Value Problems20072007:080929

Received: 5 December 2006

Accepted: 3 May 2007

Published: 27 May 2007


A generalization of the comparison principle for a semilinear and a quasilinear parabolic equations with nonlocal boundary conditions including changing sign kernels is obtained. This generalization uses a positivity result obtained here for a parabolic problem with nonlocal boundary conditions.


Boundary ConditionDifferential EquationPositivity ResultPartial Differential EquationOrdinary Differential Equation


Authors’ Affiliations

Department of Mathematics, Shanghai University, Shanghai, China
Institut für Mathematik, Universität Zürich, Zürich, Switzerland
Department of Mathematics, Campus Universitaire, University of Tunis, Elmanar, Tunisia


  1. 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
  2. Pao CV: Dynamics of reaction-diffusion equations with nonlocal boundary conditions. Quarterly of Applied Mathematics 1995,53(1):173-186.MATHMathSciNetGoogle Scholar
  3. Wang Y: Weak solutions for nonlocal boundary value problems with low regularity data. Nonlinear Analysis: Theory, Methods & Applications 2007,67(1):103-125. 10.1016/ ArticleGoogle Scholar
  4. Friedman A: Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Quarterly of Applied Mathematics 1986,44(3):401-407.MATHMathSciNetGoogle Scholar
  5. Wang Y, Zhou C: Contraction operators and nonlocal problems. Communications in Applied Analysis 2001,5(1):31-37.MATHMathSciNetGoogle Scholar
  6. Yin YF: On nonlinear parabolic equations with nonlocal boundary condition. Journal of Mathematical Analysis and Applications 1994,185(1):161-174. 10.1006/jmaa.1994.1239MATHMathSciNetView ArticleGoogle Scholar
  7. Wang Y, Nai B: Existence of solutions to nonlocal boundary problems for parabolic equations. Chinese Annals of Mathematics. Series A 1999,20(3):323-332.MATHMathSciNetGoogle Scholar
  8. Wang Y: Solutions to nonlinear elliptic equations with a nonlocal boundary condition. Electronic Journal of Differential Equations 2002,2002(5):1-16.Google Scholar
  9. Wang R-N, Xiao T-J, Liang J: A comparison principle for nonlocal coupled systems of fully nonlinear parabolic equations. Applied Mathematics Letters 2006,19(11):1272-1277. 10.1016/j.aml.2006.01.012MATHMathSciNetView ArticleGoogle Scholar
  10. Carl S, Lakshmikantham V: Generalized quasilinearization method for reaction diffusion equations under nonlinear and nonlocal flux conditions. Journal of Mathematical Analysis and Applications 2002,271(1):182-205. 10.1016/S0022-247X(02)00114-2MATHMathSciNetView ArticleGoogle Scholar


© Y.Wang and H. Zorgati. 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.