What exactly happens if you send a BTC tx with a fee of $0.50

BCH is great because there is no stupid Block Size Cap

I understand that in BTC is there a blocksize cap and if i were to send a transaction with a fee of $0.50 - The miners will reject it

But what exactly happens to that transaction?

Does it just stay unconfirmed for ever? and is stuck and remains in the mempool for ever?

What happens if 100,000 people send BTC transactions with a fee of 0.50$? Do all of those tx remain in the mempool