The expectation of a branching diffusion process with continuous time

We consider a branching diffusion process in a bounded domain with absorbing boundary. For the asymptotic behaviour of the mathematical expectation of this process we prove that
$$ M_tf(x)=e^{\mu_0t}\omega_0(x)\omega_0^*(f)+O(e^{\rho t}),\qquad t\to\infty, $$
where $M_t$ is a corresponding semigroup, $\mu_0$, $\omega_0(\cdot)$, $\omega_0^*(\cdot)$ are the first eigenvalue and the first eigenvector of the infinitesimal (adjoint) operator respectively. The proof is based on the representation of the semigroup by means of the corresponding infinitesimal operator.