Local Null Controllability Of A Free-boundary Problem For The Semilinear 1D Heat Equation

We investigate in this paper locall null controllability of a free boundary problem for the semilinear 1D heat equation:T>0,0<a<b<L_*<L₀,the initial y0?C^2+??[0,L₀]?are given,exist v?C^?,?/2???,Find L?C^1+?[0,T])????0,1/2??with 0<L_*? L?t?? B t ??0,T??0.4?Consider the equation as follows:with the free boundary condition L'?t?=-yx?L?t?,t?,t??0,T?.?0.6?Here QL = {?x,t?:x ??0,L?t??,t ??0,T?}.y =y?x,t?is the state,v = v?x,t?is a control;it acts on the system at any time through the nonempty open set ?=?a,b?,0<a<b<L_* I_? denotes the characteristic function of the set ?.The main method of this paper is to use locall null controllability results for the linear heat equation in a non-cylindrical domain and the regularity property of the solution,infer to carleman estimate of corresponding dual system and an observability inequality,finally find free-boundary and triplets by using fixed point theorem,thus come tue that the system is locall null controllability at time t = T.The main conclusion is as follows:Assume that f ? C1?R×R?,f' is Lipschitz continuous and f?0,0?= 0.Also,assume that T>0,B>0 satisfy 0<ab<L_*<L₀<B.Then?0.4?-?0.6?is locally null-controllable.More precisely,there exists ?>0,such that if ?y⁰?C^2+??[0,L₀]???,then y?x,T?=0 x??0,L?T??.