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-(xαux)+,u(x,t)ux+p(x,t,u)=h(x,t)χω,(x,t)∈ QT,u(0,t)=u(1,t)=0,t∈(0,T),u(x,0)=u0(x),x∈(0,1),where 0<α<1,T>0,QT-(0,1)x(0,T),μ∈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,χω is the characteristic function of ω=(a,b)with 0<a<b<1,u0{x)∈L2((0,1)).Based on the following assumptions:(ⅰ)reaction term p(·,·,u)is Lipschitz continuous with respect to u∈R,which is(ⅱ)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<α<1.At first,we establish the well-posedness of the original problem,then we get the regularized problem,in the case of α∈(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.