Null Controllability For The Degenerate Parabolic Equation With A Nonlinear Reaction Term

We consider the null controllability for the one-dimensional degenerate parabolic equa-tion with a nonlinear reaction term,problem is as follows:ut-(x?ux)+,u(x,t)ux+p(x,t,u)=h(x,t)??,(x,t)? QT,u(0,t)=u(1,t)=0,t?(0,T),u(x,0)=u0(x),x?(0,1),where 0<?<1,T>0,QT-(0,1)x(0,T),??L?(0,T;W1,?(0,1)),p is a measurable function on(0,1)x(0,T)x R,and it is nonlinear with respect to u?R,h is the control function,?? is the characteristic function of ?=(a,b)with 0<a<b<1,u0{x)?L2((0,1)).Based on the following assumptions:(?)reaction term p(·,·,u)is Lipschitz continuous with respect to u?R,which is(?)reaction term p?C((0,1)×(0,T)× R)satisfies p(x,t,u)|? P(|u|)|u|,(x,t)?(0,1)×(0,T),u?R,where P?C([0,+?])is an increasing function satisfying we consider the null controllability of the problem when 0<?<1.At first,we establish the well-posedness of the original problem,then we get the regularized problem,in the case of ??(0,1),we establish the Carleman estimate and observability inequalities for the linear conjugate problem;subsequently,we investigate the approximate controllability of the regularized problem by using observability inequalities and the Kakutani fixed point theorem;at last,owing to some priori estimates and the method of taking limit,we obtain the null controllability of the system.