2020
DOI: 10.1515/jiip-2017-0082
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Conditional stability in a backward Cahn–Hilliard equation via a Carleman estimate

Abstract: We consider a Cahn–Hilliard equation in a bounded domain Ω in {\mathbb{R}^{n}} over a time interval {(0,T)} and discuss the backward problem in time of determining intermediate data {u(x,\theta)}, {\theta\in(0,T)}, {x\in\Omega} from the measurement of the final data {u(x,T)}, {x\in\Omega}. Under suitable a priori boundness assumptions on the solutions {u(x,t)}, we prove a conditional stability estimate for the semilinear Cahn–Hilliard equation\lVert u(\,\cdot\,,\theta)\rVert_{L^{2}(\Omega)}\leq C\lVert u(\,\cd… Show more

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Cited by 3 publications
(2 citation statements)
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“…As a remark, the discussion of the continuous dependence is delicate for backward problems in general. In such a problem, under the assumption of the existence of bounded solutions, the conditional stability is discussed in some sense in [28] (see references therein) and in [46] for the Cahn-Hilliard equation.…”
Section: Main Theoremsmentioning
confidence: 99%
“…As a remark, the discussion of the continuous dependence is delicate for backward problems in general. In such a problem, under the assumption of the existence of bounded solutions, the conditional stability is discussed in some sense in [28] (see references therein) and in [46] for the Cahn-Hilliard equation.…”
Section: Main Theoremsmentioning
confidence: 99%
“…where Ω is a bounded domain in R n , describing the process of isothermal phase separation in a binary alloy of fluids [8] was recently considered in [30]. We finally mention the fourth-order nonlinear PDE…”
Section: Introductionmentioning
confidence: 99%