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 $\sigma (t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation ${\mathrm{\partial }}_{t}u-{\mathrm{\Delta }}_{x}u+\sigma (t)f(x)u=0$, 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 ${\mathrm{\partial }}_{t}u-{\mathrm{\Delta }}_{x}u=F(t,x,\sigma (t),u(x,t))$.