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