TY - JOUR

T1 - Stability of degenerate heat equation in non-cylindrical/cylindrical domain

AU - Gao, Hang

AU - Li, Lingfei

AU - Liu, Zhuangyi

N1 - Publisher Copyright:
© 2019, Springer Nature Switzerland AG.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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.

AB - 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.

KW - Degenerate heat equation

KW - Non-cylindrical domain

KW - Stability

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U2 - 10.1007/s00033-019-1166-3

DO - 10.1007/s00033-019-1166-3

M3 - Article

AN - SCOPUS:85069000711

SN - 0044-2275

VL - 70

JO - Zeitschrift fur Angewandte Mathematik und Physik

JF - Zeitschrift fur Angewandte Mathematik und Physik

IS - 4

M1 - 120

ER -