Carleman Estimates And Null Controllability For Degenerate Parabolic Equations With First Order Terms

This paper studies the Carleman estimates and null controllability for degenerate parabolic equations with first order terms.The equations are as follows ut-(a(x)ux)x+(b(x,t)u)x+c(x,t)u=hχω,(0,1)∈(0,T),u(0,t)=u(1,t)=0,t ∈(0,T),u(x,0)=u0(x),x ∈(0,1),where b ∈ W∞2,1((0,1)×(0,T)),c ∈ L∞((0,1)×(0,T)),h is the control function,χωis the characteristic function of ω=(x0,x1)with 0<x0<x1<1,u0∈L2((0,1)),a satisfiesⅰ)0<a ∈C2((0,1])∩ C([0,1])and a(0)=0;ⅱ)There exists some K ∈[0,1)and λ<K,such that λa(x)<xa’(x)<Ka(x)holds for x∈[0,1];ⅲ(?)ⅳ)There exists some C0>0 such that |(a(x)-xa’(x)/a(x))’|<C0a(x)/x2 holds for x∈(0,1].The above equations are degenerate at the lateral boundary,and the first order terms in the equations cannot be controlled by diffusion terms.This paper first establishes the uniform Carleman estimates and the observable inequalities for the dual problems.Then,using the fixed point theorem,the Carleman estimates and the observable inequalities obtained above,it is proved that this kind of parabolic problems is null controllable.