relationship to other variables quantified before trying to make blanket statements about "the instrument". I'd suggest looking at a text like "Physics and the Sound of Music" by Rigden or something non mathematical (assuming you are a musician and not a scientist/engineer/etc) by Fletcher and Rossing.
As a final note I will say this. The hammer placement on a piano means that you will likely excite higher pitch harmonics with each note. This is the opposite situation as my guitar example where I imaging thumbing it on center (like Wes Montgomery). In such cases the lower strings will have a chance to excite many more other strings in the harp, each at the higher harmonic. Again using the guitar example, if I play the low E string but pick it near the bridge I will excite the open string B string (n = 3) and the open high E (n = 4). These will vibrate in their fundamental mode of vibration as those frequencies match the higher harmonics of the low E. NOTE: plucking near the bridge is critical for this to work well. So it is possible that lower stings in the piano have several octaves of strings helping to support the harmonics. But again, as the string motion dies I question if it is the low pitch fundamental that you hear or the ringing of all the harmonics. It would be natural to associate this ringing with the string you hit but that isn't necessarily true. It may be all the others. This in no way contradicts the previous example but serves to illustrate the complexity of the instrument and that with the correct conditions either phenomenon can be observed.
This is a really interesting and complicated question in the physical simulation of string dynamics.
Actually, it’s not entirely true that high pitch notes run shorter. There is a tendency for higher order partials (inharmonic overtones) to decay faster (run shorter). But due to the complexities of piano tuning and string coupling, it is not true that if you play each note on a piano one after another, the next higher note will always decay faster. Also, you will find that each partial within a particular note might decay in a different manner, and this pattern of decay varies per note.
Tackling the simpler part of the answer, one of the reasons there’s a tendency for higher order partials to decay faster can be seen in a simplified model of string vibrations.
The wave equation of a simple oscillating string, with decay, can be modeled with the formula.
The gist of the formula is: the bit on the left, ψ(x,t), determines the displacement of the string at some point x and some time t. The goal of this wave equation is to determine where every point will be at every time, which is the same as knowing vibration frequency and amplitude. The stuff inside the cosine on the right can be ignored for this discussion. But the key important thing is the exponential e-κx. κ is proportional to the frequency of the wave. Which means as frequency goes up, κ goes up, which means the exponent shrinks, and so the physical displacement of the string from rest is decreased. In other words, there’s decay dependent on frequency.
This isn’t the full story, but does indicate that high frequencies lose energy to the air faster. There are other sources of energy decay though, like loss in the sound board of the instrument.
Other complexities arise, like the coupling between strings. Take an A4, which has three strings. If all three strings are tuned to the exact same frequency, the decay rate will be three times faster than a single string. However in practice the strings are tuned very slightly differently, creating beats in higher order partials. If one partial decays faster than others, because of coupling, it transfers energy into the others. Thus the note is sustained much longer. As you move along the piano, from lowest to highest, some notes have one string, some two and the majority have three. There will be kinks in the decay pattern as you transition across these regions.
The diagram below comes from the paper http://matthiasmauch.de/_pdf/cheng2015modelling.pdf and does a good job 