Identification Of Parameters In Periodic Parabolic Equations

Let be an open and bounded domain with C2-boundary and be a subdomain. Let . From now on, we shall omit all x and t in the functions of x, t, if there is no ambiguity. We consider the following linear parabolic equationwhere we denote yt or y' the derivative of y(x, t) tot, f L2(Q) is a given function. The function a(x) is taken from a certain set K of functions which will be precised later.The purpose of this work is to identify a(x) from set K via the observations on subdomain u C 0. More precisely, we shall study the following identification problem: (P) inf L(y, a) over all (y, a) H2,1(Q) x K satisfying (1.1), where is a given function, c1. and c2 are constants.The key step to get the existence of the solutions for (P) here is to use the Carleman inequality. With the help of the Carleman inequality and the periodic condition, we get the boundness of a minimizing sequence {yn} to problem (P) in a suitable space. Then by Ascoli-Arzela Theorem and Aubin Lemma, we derive the existence of the solutions to problem (P).