Approximate Controllability Of A Class Of Semilinear Parabolic Systems With Boundary Degeneracy

This paper concerns the approximate controllability of the following semilinear parabolic system with boundary degeneracy:(?)(x,y,t)? QT,u(x,y,t)= 0,(x,y,t)??,u(x,y,0)= u0(x,y),(x,y)? G,where QT=(0,1)×?×(0,T),G =(0,1)×?,?= {(x,y,t)?(?)G ×(0,T):a(x,y,t)>0} ? {(x,y,t)?(?)G ×(0,T):a(x,y,t)= 0 and (?)The coefficient ??C((?)T)?C1(QT)and is positive on G×[0,T],aij?C((?)T)?C1(QT)(i,j = 1,2...n)satisfy (?)aij(x,y,t)= aji(x,y,t),(x,y,t)? QT,i,j = 1,2,...n,(?) with 0<?<? and M>0 be some constants.That is,only the coefficient a leads to the weak degeneracy or non-degeneracy on the lateral boundary of the system.This paper is divided into five chapters:Chapter 1 is introduction,where the developmen-t for approximate controllability of parabolic system with boundary degeneracy and the previous results are introduced,and the difference from other papers is presented.Chapter 2 is preliminaries,in which we state the definition of weak solutions,and es-tiablish the well-posedness of weak solutions and some compact estimates by means of the parabolic regularized method as well as the Holmgren's method.Chapter 3 shows the well-posedness of semilinear parabolic system by Schauder's fixed point theorem and the well-posedness of linear parabolic system.Chapter 4 concerns the existence of weak solutions of conjugate problems and proves the approximate controllability of linear parabolic system by constructing the control function which is relateds to solu-tions of conjugate problems.Chapter 5 demonstrates the approximate controllability of semilinear parabolic system by the Kakutani's fixed point theorem.