Stability for an inverse source problem of the diffusion equation

We prove an increasing stability estimate for an inverse source problem of the diffusion equation with a nontrivial variable absorption coefficient in R3. The stability estimate consists of two parts: the Lipschitz type data discrepancy and the high frequency tail of the source function. The latter decreases when the upper bound of the frequency increases. The proof is based on an exact observability bound for the heat equation and the resolvent estimates for the elliptic operator. In particular, to justify the Fourier transform and obtain Plancherel's theorem for the time-domain Cauchy data on the boundary, we derive certain appropriate time decay estimates for the heat equation using semigroup theory.