Stability of the determination of a time-dependent coefficient in parabolic equations

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We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient σ(t), appearing in a Dirichlet initial-boundary value problem for the parabolic equation ∂ₜu-Δₓ u+σ(t)f(x)u=0, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation in changed to the semi-linear parabolic equation ∂ₜu-Δₓ u=F(t,x,σ(t),u(x,t)).