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Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Discontinuous Parabolic Problems with a Nonlocal Initial Condition | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 965759 10 pages doi 10.1155 2011 965759 Research Article Discontinuous Parabolic Problems with a Nonlocal Initial Condition Abdelkader Boucherif Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals P.O. Box 5046 Dhahran 31261 Saudi Arabia Correspondence should be addressed to Abdelkader Boucherif aboucher@kfupm.edu.sa Received 28 February 2010 Revised 31 May 2010 Accepted 13 June 2010 Academic Editor Feliz Manuel Minhos Copyright 2011 Abdelkader Boucherif. 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. We study parabolic differential equations with a discontinuous nonlinearity and subjected to a nonlocal initial condition. We are concerned with the existence of solutions in the weak sense. Our technique is based on the Green s function integral representation of solutions the method of upper and lower solutions and fixed point theorems for multivalued operators. 1. Introduction Let Q be a an open bounded domain in RN N 2 with a smooth boundary dQ. Let QT Q X 0 T and rT dQ X 0 T where T is a positive real number. Then rT is smooth and any point on rT satisfies the inside and outside strong sphere property see 1 . For u QT R we denote its partial derivatives in the distributional sense when they exist by Dtu du df Diu du dxi DiDjU Ố2u dxịdxj i j 1 . N. In this paper we study the following parabolic differential equation with a nonlocal initial condition Dtu Lu f x f ũ x f e Qt u x f 0 x f e rT 1.1 c T u x 0 k x f u x ff df x e Q 0 2 Boundary Value Problems where f QT X R R is not necessarily continuous but is such that for every fixed u e R the function x f f x t u is measurable and u f x t ù is of bounded variations over compact interval in R and nondecreasing and k Qt X R R is continuous L is
