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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On Comparison Principles for Parabolic Equations with Nonlocal Boundary Conditions | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 80929 10 pages doi 10.1155 2007 80929 Research Article On Comparison Principles for Parabolic Equations with Nonlocal Boundary Conditions Yuandi Wang and Hamdi Zorgati Received 5 December 2006 Revised 8 March 2007 Accepted 3 May 2007 Recommended by Peter Bates 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. Copyright 2007 Y. Wang and H. Zorgati. 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. 1. Introduction The positivity of solutions for parabolic problems is the base of comparison principle which is important in monotonic methods used for these problems. Recently Yin 1 developed several results in applications of the comparison principle especially on nonlocal problems. Earlier works on problems with nonlocal boundary conditions can be found in 2 and some of references can be found in 1 3 . In the literature for example 2 4-6 a restriction on the boundary condition see 2.1 of the kind k x y dy 1 k x y 0 AK n where k represents the kernel of the nonlocal boundary condition is sufficient to obtain the comparison principles. Recent results show that this restriction is not necessary for problems with lower regularity see 3 Theorem 3.11 for problem with Dirichlet-type nonlocal boundary value . Moreover in 7 an existence result for classical solutions of a parabolic problem with nonlocal boundary condition was obtained. In 8 we find an illustration of how the boundary kernel influences some results such as those on the eigenvalues problem and on the decay of solutions for evolution .