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

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 $\partial_tu-Δ_x u+σ(t)f(x)u=0$, 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 $\partial_tu-Δ_x u=F(t,x,σ(t),u(x,t))$.

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