  We consider the reconstruction of the diffusion and absorption coefficients
of the diffusion equation from the internal information of the solution
obtained from the first step of the inverse photoacoustic tomography (PAT). In
practice, the internal information is only partially provided near the boundary
due to the high absorption property of the medium and the limitation of the
equipment. Our main contribution is to prove a H\"older stability of the
inverse problem in a subregion where the internal information is reliably
provided based on the stability estimation of a Cauchy problem satisfied by the
diffusion coefficient. The exponent of the H\"older stability converges to a
positive constant independent of the subregion as the subregion contracts
towards the boundary. Numerical experiments demonstrates that it is possible to
locally reconstruct the diffusion and absorption coefficients for smooth and
even discontinuous media.
