The Stability Of A One-dimensional Heat Equation With Variable Coefficient

In this paper we mainly consider the following system of one-dimensional heat equation with variable coefficient.here Q= {(x,t)|x∈(0,1),t∈(0,∞)},where k is a positive constant.First we use the energy method to prove the solution of system is polynomial stable,and we use separation of variables to obtain Fourier type series solution for the system,the solution can only be polynomial decay to zero rather than rapid exponential decay by the estimate for solution of the system.we use variable substitution to obtain an equivalent system,then we construct a boundary feedback control,and the control acts on the right-side moving boundary of the equivalent system,the rapid exponential stabilization is obtained by the backstepping method,then the original system is also exponential stable.