No, my heart rate goes up (like everyone else's) but doesn't come back down for a while (10 minutes?). I have no endurance and resting between workouts doesn't recharge me.
Obviously, Your two sentences is probably not enough for me to help you, but I'll give it a shot anyway. Having a high heart rate that you can sustain is, in my experience, actually good for endurance sports. In fact, my father pointed out the other day that he just can't get his heart rate up any more (he's 73 and still rides his bike over mountains). This means he can't get enough oxygen into his muscles to keep a high pace. As I've grown older, I've also noticed this ceiling that is limited by how high I can sustain my heart rate and for how long.
So, rather than assume that your observation is the problem, I think perhaps you might have something you can work with, whatever your underlying problem actually is. I've done sports my whole life (running, a small bit of weight lifting, karate, and now cycling). Especially now that I'm getting older, I need to concentrate on "base miles" on my bike. This means going out at a very slow pace (annoyingly so, for me) and riding for very long time. Cycling is good for this as it is relatively easy to dial the pace to whatever you need -- just make sure to have a lot of low gears if you tend to have problems.
I suspect that if you ride a bike at a very slow pace for an hour a day for about 6 months that you will find a really marked difference in your ability to exercise. Keep your pace so that you could have a conversation easily and that you aren't out of breath. If you run into difficult situations (hills, wind, whatever), just go into a lower gear. You may need to go to a bike shop to get especially low gears, but it won't be outrageously expensive.
Hope that helps.
Having a good sleep and some sugar before improves the condition.