Asymptotic behaviour of a transport equation

We study the asymptotic behaviour of the semigroup of Markov operators generated by the equation ${\displaystyle u_t+b{u}_x+cu=a{\int }_0^{ax}u(t,ax-y)\mu \left(dy\right)}$. We prove that for a > 1 this semigroup is asymptotically stable. We show that for a ≤ 1 this semigroup, properly normalized, converges to a limit which depends only on a.