Quantitative unique continuation for the semilinear heat equation in a convex domain |
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Authors: | Kim Dang Phung Gengsheng Wang |
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Affiliation: | a Yangtze Center of Mathematics, Sichuan University, Chengdu 610065, China b School of Mathematics and Statistics of Wuhan University, Wuhan 430072, China |
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Abstract: | In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation t∂u−△u=g(u), with the homogeneous Dirichlet boundary condition, over Ω×(0,T∗). Ω is a bounded, convex open subset of Rd, with a smooth boundary for the subset. The function g:R→R satisfies certain conditions. We establish some observation estimates for (u−v), where u and v are two solutions to the above-mentioned equation. The observation is made over ω×{T}, where ω is any non-empty open subset of Ω, and T is a positive number such that both u and v exist on the interval [0,T]. At least two results can be derived from these estimates: (i) if ‖(u−v)(⋅,T)‖L2(ω)=δ, then ‖(u−v)(⋅,T)‖L2(Ω)?Cδα where constants C>0 and α∈(0,1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over ω×{T}, then they coincide over Ω×[0,Tm). Tm indicates the maximum number such that these two solutions exist on [0,Tm). |
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Keywords: | Semilinear heat equation Quantitative unique continuation |
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