We are concerned with the stability of viscosity solutions to contact Hamilton-Jacobi equation
H
(
x
,
∂
x
u
(
x
)
,
u
(
x
)
)
=
0
,
x
∈
M
,
\begin{align*} H(x,\partial _x u(x),u(x))=0, \quad x\in M, \end{align*}
where
H
=
H
(
x
,
p
,
u
)
H=H(x,p,u)
satisfies Tonelli conditions. We study the relationship between Lyapunov stability of viscosity solutions and the structure of the set of weak KAM solutions to the contact Hamilton-Jacobi equation.