Gravitational wave and 1st law of thermodynamics Introduction:
A prediction of the general relativity is that any moving mass produces fluctuation in the space-time fabric, commonly referred as Gravitational-Wave.
This prediction was recently confirmed by the LIGO experiment.
The generation of such gravitational waves requires energy, as stated on the wiki article linked above:

Water waves, sound waves, and electromagnetic waves are able to carry
energy, momentum, and angular momentum and by doing so they carry
those away from the source. Gravitational waves perform the same
function. Thus, for example, a binary system loses angular momentum as
the two orbiting objects spiral towards each other—the angular
momentum is radiated away by gravitational waves.

The first law of thermodynamics states that:

The first law of thermodynamics, also known as Law of Conservation of
Energy, states that energy can neither be created nor destroyed

Given that, one can imply that any moving object having a mass would create gravitational waves - even ever so tiny -, thus having a drag.
Question:
How does a system, for instance earth-moon orbit, can be stable and not decaying over time over the model of the general relativity? (Where does the energy comes from?)
Is this question solved?
 A: 
any moving object having a mass would create gravitational waves

This is not true in general. At the lowest order, the rate of energy lost as gravitational waves is proportional to the third time derivative of the quadrupole moment tensor of the system. For example, a mass moving at constant velocity will not radiate gravitational waves. Neither will a spherically symmetric mass distribution.

How does a system, for instance earth-moon orbit, can be stable and not decaying over time over the model of the general relativity? (Where does the energy comes from?)

The Earth-Moon system is a binary system, and the quadrupole moment tensor of such a system will have a non-zero third time derivative. As such, it is losing energy through gravitational waves and consequently decaying.

Is this question solved?

Yes, at least approximately. To lowest order, the rate of decay of a binary system with masses $m_1$ and $m_2$ is
$$\frac{\mathrm{d}r}{\mathrm{d}t} = - \frac{64}{5}\, \frac{G^3}{c^5}\, \frac{(m_1m_2)(m_1+m_2)}{r^3}$$
We can already see from the factor $G^3/c^5$ that the rate of decay is extremely small. Plugging in typical solar system values, we can see that the time needed for any significant decay is much longer than the age of the universe.
A: The objects that are radiating gravitational waves must be losing the energy and angular momentum they had in their orbits. For example, when two black holes spiral in toward each other, they radiate away this energy and angular momentum as their orbit decays and they eventually merge.

How does a system, for instance earth-moon orbit, can be stable and not decaying over time over the model of the general relativity?

For the moon-Earth system, the amount of gravitational wave radiation is incredibly small. The time needed for any noticeable orbital decay is greater than the age of the universe. More energy is lost to dissipative forces, and so the moon or planets are in fixed orbits where the total energy and angular momentum is practically constant.
A: The energy comes from the binary system itself and ultimately from the mass-energy of the binary components.
The binding energy of a binary system (the sum of its kinetic and potential energies) is negative. The acceleration of these masses, or more precisely, the accelerating mass quadrupole moment, produces gravitational waves that carry away energy and angular momentum. The binary components move closer together and the system binding energy becomes more negative and thus total energy is conserved.
This is happening in the Earth-Moon system but the flux of gravitational waves, which depends roughly on mass to the power of 5 and inversely on the component separation to the power of 5, is pitifully weak (about $10^{-5}$W) compared to a binary black hole system. The evolution of the Earth-Moon system is instead governed by the interplay of orbital and rotational energies caused by tidal forces.
