I added a utility sink which is on the other side of the wall from my double bowl kitchen sink. The utility sink drain is connected to the kitchen sink drain pipe just above the J-trap. The utility sink backs up constantly when there is heavy use in the kitchen sink/disposal. I can't figure out why. All the food waste from the disposal ends up in the utility sink. What did I do wrong? Is there a solution?

The utility sink backs up constantly when there is heavy use in the kitchen sink/disposal. You're overloading the kitchen drain line, especially if you have a older place with a 1 1/2" drain line.
You system can't handle the volume you're putting into it.
A possible solution might be to snake the kitchen roof vent. Put out enough snake to reach the base and 20 feet more. Good luck, Tom

Two sinks on one trap and the utility sink bottom is probably 6" lower than the kitchen sink. Water routes itself to the least resistance. I'd consider trapping the uttility sink and then joining the drain after the kitchen trap.