I am not sure what you mean exactly, but here are some suggestions:
- There are terms like “morando”, or “mancando”, meaning dying/dying away. You could also literally write out “exhaustion” or the various derivatives and synonyms of that word (e.g. “expiring”). I suppose they’re rather similar to rallentando/diminuendo so just a change in semantics/terminology is probably not what you’re looking for.
- Manipulate accents and note values. (1) More accented notes could give the feeling of more effort and energy (one might indicate “con forza” or simply with sforzandi and similar accents); and perhaps change it near the end, such as by playing notes with less effort, as if exhausted. (2) Hold notes for a longer time (one might indicate “tenuto”) and then maybe change to staccato or half the note value near the end, which also helps with the feeling of less effort, of “expiring away”.
P.S. In regards to note values, one specific way of doing it might be as follows. If you want to keep a constant tempo (rhythmic, on the beat) and perhaps even roughly equal volume, but still give some semblance of “running out of steam”, one way I can think of is to assign increasing/decreasing note values. For example, like a train gaining steam and slowing down, one might use quarter notes first, then 8th notes, then triplet 8ths, then 16ths, 32nd, and trill, and then perhaps decrease the note value after the trill peak or simply return right away to longer note values like quarter notes, half notes, etc. near the end. I suppose usually such passages are still phrased with changes in tempo and volume, but I think even at a constant tempo or volume this can produce an effect of build-up and exhaustion. If you’re looking for a specific example of such an instance, one that I particularly love is that of the 6th variation of the third movement of Beethoven’s Sonata No. 30, Op. 109. At the end after the trill, he returns to the opening theme and it ends serenely. (If you haven’t listened to this sonata before, I would personally recommend listening to it from the beginning and going all the way through, but it is quite long, like 20 minutes, and I understand if you may not have the time or interest.) When you say “exhaustion”, I kind of assume there should be some sort of build-up or at least loudness and energy to exhaust from, but I might be wrong.
Sorry, I spoke essentially from a classical piano music perspective; I’m not sure if that is relevant or helps at all.