Blossom end rot. Not much you can do. It is believed to be a calcium deficiency that causes it, but who knows and it doesn't mean the compost doesn't have calcium in it either.

Try keeping the compost evenly moist and not letting it get soggy or bone dry as best you can and see if things turn around.

Also, you can do a google search on blossom end rot to get other ideas on what might help.

It's rather unusual to use only compost as the growing media. Is there any reason why you are not using potting soil or potting soil mixed with compost?

As has been said it's blossom end rot, caused by a lack of calcium...but not necessarily a lack in the soil, but in the plant's ability to take the calcium up.
Be very careful never to let the container dry out....it must be kept very much on the moist side.
And just for good measure, dry some egg shells in the oven ( to get rid of any bits of egg that might draw "critters") and crush it up and add to the soil.
Container grown tomatoes are very subject to blossom end rot....
Linda C

I recall reading an article that said BER was more common among stressed plants which had endured uneven watering - alternating drought/drown extremes, which happens easily in containers. I agree with Linda that adding crushed eggshells will likely be of help (I use shells from a dozen eggs per tomato plant). I'd also bury the liner from a couple overnight diapers around the edge of the pot (liner facing plant, plastic outer against the pot side); didies will absorb excess water and then release it gradually as the soil dries which will help counteract the drought/drown cycle.