Stability of degenerate heat equation in non-cylindrical/cylindrical domain

In this paper, we investigate the stability of a degenerate heat equation ut(x,t)=(xαux(x,t))x,x∈(0,1),t>0in a non-cylindrical/cylindrical domain. It is well known that the heat equation without degeneracy is exponentially stable in cylindrical domain. In the case of degeneracy, we first extend the existing result [25] on uniform exponential stability in cylindrical domain from α∈ (0 , 1) to α∈ (0 , 2 ]. For a class of non-cylindrical domain (linear moving boundary), we show that the stability depends on the degeneration index α. More precisely, it is not exponentially stable for α= 0 , 1 but polynomially stable for α= 1 , is analogously exponentially stable for 1 < α< 2 , and is exponentially stable for α= 2. It is interesting to see that there is a positive impact of the degeneracy on stability of the system in non-cylindrical domain.