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 ??(p,q),and 0 ?p?q ?L*?L0?B,1? 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 ?>0,such that if y~0(·)satisfies||y~0||C2+?([0,L0])??,there will exist v(·,·)? L2(?×(0,T)),then y(x,T)=0,x ?(0,L(T)).(0.4)...