Affiliation:
1. Mathematics & Science College , Shanghai Normal University , 100 Guilin Road , Shanghai 200234 , P. R. China
2. School of Mathematical Sciences , University of Science and Technology of China , 96 Jinzhai Road Baohe District , Hefei , Anhui 230026 , P. R. China
Abstract
Abstract
We consider a Cahn–Hilliard equation in a bounded domain Ω in
ℝ
n
{\mathbb{R}^{n}}
over a time interval
(
0
,
T
)
{(0,T)}
and discuss the backward problem in time of determining intermediate data
u
(
x
,
θ
)
{u(x,\theta)}
,
θ
∈
(
0
,
T
)
{\theta\in(0,T)}
,
x
∈
Ω
{x\in\Omega}
from the measurement of the final data
u
(
x
,
T
)
{u(x,T)}
,
x
∈
Ω
{x\in\Omega}
. Under suitable a priori boundness assumptions on the solutions
u
(
x
,
t
)
{u(x,t)}
, we prove a conditional stability estimate for the semilinear Cahn–Hilliard equation
∥
u
(
⋅
,
θ
)
∥
L
2
(
Ω
)
≤
C
∥
u
(
⋅
,
T
)
∥
H
2
(
Ω
)
κ
0
,
\lVert u(\,\cdot\,,\theta)\rVert_{L^{2}(\Omega)}\leq C\lVert u(\,\cdot\,,T)%
\rVert_{H^{2}(\Omega)}^{\kappa_{0}},
and a conditional stability estimate for the linear Cahn–Hilliard equation
∥
u
(
⋅
,
θ
)
∥
H
β
(
Ω
)
≤
C
∥
u
(
⋅
,
T
)
∥
H
2
(
Ω
)
κ
1
,
\lVert u(\,\cdot\,,\theta)\rVert_{H^{\beta}(\Omega)}\leq C\lVert u(\,\cdot\,,T%
)\rVert_{H^{2}(\Omega)}^{\kappa_{1}},
where
θ
∈
(
0
,
T
)
{\theta\in(0,T)}
,
β
∈
(
0
,
4
)
{\beta\in(0,4)}
and
κ
0
,
κ
1
∈
(
0
,
1
)
{\kappa_{0},\kappa_{1}\in(0,1)}
. The proof is based on a Carleman estimate with the weight function
e
2
s
e
λ
t
{\mathrm{e}^{2s\mathrm{e}^{\lambda t}}}
with large parameters
s
,
λ
∈
ℝ
+
{s,\lambda\in\mathbb{R}^{+}}
.
Funder
National Natural Science Foundation of China
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