Carleman Inequality Of Backward Linear Parabolic Equations On Non-cylindrical Domains And Local Null Controllability For Stefan Problem

In this paper,the local null controllability of a free-boundary problem(0.1)-(0.2)for the semilinear parabolic equation is studied:(?) L’(t)=-yx(L(t),t),t ∈(0,T),(0.2)Where let T>0,L0＞ 0,B＞0 be given,and L0 ＜B.The Free boundary L(t)is unknown,QL={(x,t)|x∈(0,L(t)),t∈(0,T)}.y＝y(x,t)is the state of the system,v=v(x,t)is the control function.It acts on the whole system through the noempty open set ω＝(p,q),and 0 ＜p＜q ＜L*＜L0＜B,1ω represents the characteristic function of the set w.Let us assume that y~0,and g can satisfy certain conditions.And impose restrictions on the free boundary:0<Lo≤L(t)≤B,t ∈[0,T],(0.3)In this paper,we first obtain a backward Carleman inequality for the linear parabolic equation on a non cylindrical domain with the fixed boundary.Secondly,it is proved that,if the initial state is small at the time T,the solution of the Stefan problem(0.1)-(0.2)is locally null controllable by using linearization,the fixed boundary method and the Kakutani fixed point theorem.In other words,there exists ε>0,such that if y~0(·)satisfies||y~0||C2+α([0,L0])≤ε,there will exist v(·,·)∈ L2(ω×(0,T)),then y(x,T)=0,x ∈(0,L(T)).(0.4)...