  We consider theoretically the transport in a one-channel spinless Luttinger
liquid with two strong impurities in the presence of dissipation. As a
difference with respect to the dissipation free case, where the two impurities
fully transmit electrons at resonance points, the dissipation prevents complete
transmission in the present situation. A rich crossover diagram for the
conductance as a function of applied voltage, temperature, dissipation
strength, Luttinger liquid parameter K and the deviation from the resonance
condition is obtained. For weak dissipation and 1/2<K<1, the conduction shows a
non-monotonic increase as a function of temperature or voltage. For strong
dissipation the conduction increases monotonically but is exponentially small.
