We consider a Cahn–Hilliard equation in a bounded domain Ω in ℝn{\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)∥L2⁢(Ω)≤C⁢∥u⁢(⋅,T)∥H2⁢(Ω)κ0,\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)∥Hβ⁢(Ω)≤C⁢∥u⁢(⋅,T)∥H2⁢(Ω)κ1,\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 κ0,κ1∈(0,1){\kappa_{0},\kappa_{1}\in(0,1)}. The proof is based on a Carleman estimate with the weight function e2⁢s⁢eλ⁢t{\mathrm{e}^{2s\mathrm{e}^{\lambda t}}} with large parameters s,λ∈ℝ+{s,\lambda\in\mathbb{R}^{+}}.

中文翻译：

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通过Carleman估计在反向Cahn-Hilliard方程中的条件稳定性

我们考虑时间间隔（0，T）{（0，T）}中的ℝn{\ mathbb {R} ^ {n}}中的有界域Ω中的Cahn–Hilliard方程，并讨论确定时的倒数问题中间数据u⁢（x，θ）{u（x，\ theta）}，θ∈（0，T）{\ theta \ in（0，T）}，x∈Ω{x \ in \ Omega}最终数据的测量u⁢（x，T）{u（x，T）}，x∈Ω{x \ in \ Omega}。在关于解u⁢（x，t）{u（x，t）}的合适先验有界假设下，我们证明了半线性Cahn-Hilliard方程∥u⁢（⋅，θ）∥L2⁢（ Ω）≤C⁢∥u⁢（⋅，T）∥H2⁢（Ω）κ0，\ lVert u（\，\ cdot \ ,, \ theta）\ rVert_ {L ^ {2}（\ Omega）} \ leq C \ lVert u（\，\ cdot \ ,, T）％\ rVert_ {H ^ {2}（\ Omega）} ^ {\ kappa_ {0}}，以及线性Cahn–Hilliard方程的条件稳定性估计∥ u⁢（⋅，θ）∥Hβ⁢（Ω）≤C⁢∥u⁢（⋅，T）∥H2⁢（Ω）κ1，\ lVert u（\，\ cdot \ ,, \ theta）\ rVert_ {H ^ {\ beta}（\ Omega）} \ leq C \ lVert u（\，\ cdot \ ,, T％）\ rVert_ {H ^ {2}（\ Omega）} ^ {\ kappa_ {1}}，其中θ∈（0，T）{\ theta \ in（0，T）}，β∈（0， 4）{\ beta \ in（0,4）}和κ0，κ1∈（0,1）{\ kappa_ {0}，\ kappa_ {1} \ in（0,1）}。该证明基于具有大参数s，λ∈the +的权重函数e2⁢s⁢eλ⁢t{\ mathrm {e} ^ {2s \ mathrm {e} ^ {\ lambda t}}}的Carleman估计{s，\ lambda \ in \ mathbb {R} ^ {+}}。