So the question in the topic.
I want to clarify whether the speed indications is maintained or go to 0.
Depends on how much blockage. If fully blocked, as in air tight, the pitot side becomes like the sealed aneroid side of an altimeter and it only reacts to changes in static pressure, indicating the difference between ambient static and the pressure trapped in the pitot side.
If the blockage is not air tight, like you usually get with bugs, somewhere between zero and actual, in proportion to the blockage. If the blockage is something that can move around, like a slug of water in the pitot line, you get wild and crazy fluctuations.
If the pitot tube gets blocked in flight, such as by icing, the airspeed indication is maintained at first.
Note that the indication is still subject to changes in static pressure as long as the static port is not blocked as well, effectively turning the airspeed indicator into an altimeter.
If only the static port is blocked, the airspeed indicator shows a weird combination of airspeed and altitude.
If the Pitot is ice over or plugged by a bug in flight, the ASI will indicate the speed at the time it was block as long as the pressure remains constant in the bourdon tube. The ASI will change with any altitude change, acting as a altimeter. If the is a climb the ASI will show an increase in speed, if descending there will be a decrees in airspeed. If the static ports a pugged or iced over the ASI will act a s altimeter in reverse.
Look at it this way:
The airspeed indicator is basically an instrument that provides the pilot with the difference between static and dynamic pressure, reported in a scale calibrated in a unit of speed (usually knots, which is nautical miles per hour; sometimes km/h or mi/h).
The static pressure is measured at the static port, and the dynamic pressure is measured through the pitot tube; in both cases, the one of each which is connected to the airspeed indicator. Very small aircraft will have one of each (unpressurized aircraft might not have a dedicated static port, but just be measuring that at or near the instrument), while large aircraft will have several of each.
When you're flying normally at a constant pressure altitude (say, holding a specific flight level), this means that the airspeed indicator reacts to changes in dynamic pressure, which in turn is basically how much air a given frontal area of the aircraft encounters per unit time. Go faster, dynamic pressure goes up, static pressure stays the same, difference increases so the airspeed indicator says your speed is increasing; go slower, dynamic pressure goes down, difference decreases, airspeed indicator says your speed is decreasing. Fine so far.
Now consider what happens if the pitot tube gets completely blocked, as in absolutely air tight on the side exposed to the ambient air. Whatever pressure there was before the blockage developed (for the sake of simplicity, let's say it got instantly and completely blocked by magic) will remain in the pitot system, but the static pressure measurement will still read ambient pressure correctly.
At that moment, for the airspeed indicator, nothing has changed; the static and dynamic pressure are both unchanged, and the aircraft is in fact still moving through the air at the same speed. So the airspeed indicator reports the same difference in pressure, ergo the same airspeed. It does not drop to zero.
Now consider what happens in such a situation when you climb or descend; in other words, when the static pressure changes.
When you descend, the static pressure increases (last I looked, air was thicker closer to the ground) but because it's blocked, the pressure in the pitot line remains unchanged. This causes the difference between the two to decrease, which as we established above is displayed by the airspeed indicator as a decrease in speed.
Correspondingly, when you climb with a blocked pitot tube, the static pressure decreases and the pitot pressure remains unchanged. Therefore the difference between the two increases, which the airspeed indicator reports as an increase in speed.
In other words, the airspeed indicator now acts as a crude altimeter! (In fact, reporting the difference between static pressure and a reference pressure is exactly what an altimeter is designed to do. It is, however, usually calibrated in a unit more useful for that purpose than knots...)
Both of these are the exact opposites of what would normally happen when you climb or descend at constant engine thrust.
If the pitot tube is only partially blocked, then some of the change in dynamic pressure will make it to the airspeed indicator, and therefore the difference will be less pronounced, but the error will still be in the same direction as if it was completely blocked at a given time.