A battle goes live only after the other owner accepts and both sides have paid the buy-in. A challenge that is declined or not answered in time moves to the expired state and never goes live. If you were declined, the inbox shows the decline message when the other owner wrote one.
If a challenge expires or is cancelled after one side has already paid the buy-in, that buy-in is refunded.
If a battle you expected is not on the Live tab, check Upcoming, My battles and Finished, and check that the chain filter matches the battle's chain.