Stability Results for Semi-linear Parabolic Equations Backward in Time

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• 作者：Nguyen Van Duc ; Nguyen Van Thang
• 关键词：Semi ; linear parabolic equations backward in time ; Ill ; posed problems ; Stability estimate ; Log ; convexity method ; Tikhonov regularization
• 刊名：Acta Mathematica Vietnamica
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：42
• 期：1
• 页码：99-111
• 全文大小：
• 刊物主题：Mathematics, general;
• 出版者：Springer Singapore
• ISSN：2315-4144
• 卷排序：42

文摘

Let H be a Hilbert space with the norm ∥⋅∥, and let A:D(A) ⊂ H → H be a positive self-adjoint unbounded linear operator on H such that −A generates a C₀ semi-group on H. Let φ be in H, E > ε a given positive number and let f : [0, T]×H → H satisfy the Lipschitz condition ∥f(t, w₁)−f(t, w₂)∥ ≤ k∥w₁−w₂∥,w₁,w₂∈H, for some non-negative constant k independent of t, w₁ and w₂. It is proved that if u₁ and u₂ are two solutions of the ill-posed semi-linear parabolic equation backward in time u_t + Au = f(t, u), 0 < t ≤ T,∥u(T)−φ∥ ≤ ε and ∥u_i(0)∥ ≤ E, i = 1,2, then$$\|u_{1}(t)-u_{2}(t)\| \leq 2\varepsilon^{t/T} E^{1-t/T}\exp\Big[\Big(2k+\frac{1}{4}k^{2}(T+t)\Big)\frac{t(T-t)}{T}\Big] \quad \forall t \in [0,T]. $$ The ill-posed problem is stabilized by a modification of Tikhonov regularization which yields an error estimate of Hölder type.KeywordsSemi-linear parabolic equations backward in timeIll-posed problemsStability estimateLog-convexity methodTikhonov regularizationMathematics Subject Classification (2010)35K58References1.Agoshkov, V.I.: Optimal Control Methods and the Method of Adjoint Equations in Problems of Mathematical Physics. Russian Academy of Sciences, Institute for Numerical Mathematics, Moscow (2003). (Russian)MATHGoogle Scholar2.Alifanov, O.M.: Inverse Heat Transfer Problems. 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Nonlinear Anal. 71, 4167–4176 (2009)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016Authors and AffiliationsNguyen Van Duc1Email authorNguyen Van Thang11.Department of MathematicsVinh UniversityVinh CityVietnam About this article CrossMark Publisher Name Springer Singapore Print ISSN 0251-4184 Online ISSN 2315-4144 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; 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