Abstract :
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).
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