Quantitative unique continuation for the semilinear heat equation in a convex domain

In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation ∂tu− 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). © 2010 Elsevier Inc. All rights reserved.