Conditional stability in a backward Cahn–Hilliard equation via a Carleman estimate
Yunxia Shang, Shumin Li · Journal of Inverse and Ill-Posed Problems · 2020
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