Quantitative Unique Continuation For The Linear Coupled Heat Equations

In this dissertation,we establish certain unique continuation results for a coupled heat equations,with the homogeneous Dirichlet boundary condition and time dependent coefficients,on a bounded convex domain ?.The dissertation consists of three parts.The organizing structure is as follows:Chapter One is divided into two sections.In Section One,we introduce the research status of the unique continuation for the partial differential equations and the linear coupled heat equations which will be studied in the paper.While in Section Two,we present the main inequalities as follows:andChapter Two,we first give the properties of the functions G?(x,t),H?(t),D?(t)and the frequency function N?(t).Then,we prove the Theorem 1.1 and Theorem 1.2 based on the skill of frequency functions.The following results may be easily seen from the theorem:1.if y(x,T)? z(x,T)= 0 on ?,then y(x,T)= z(x,T)= 0 on ? ×(0,T).2.if y(·,T)= z(·,T)= 0 on ?,then y = z = 0 on ? ×(0,T).Chapter Three is the conclusions,which reviews the main contents of the full text.In this paper,we establish and prove the certain unique continuation results for a coupled heat equations.Depending on the results of this paper,we can obtains the quantitative unique continuation for more kinds of partial differential equations.This applies even more to control theory.