In the below position, what is a clear line that shows Black escapes from perpetual check?
It seems that the black king can escape via a5:
1...Kb8 2.Qd6+ Kb7 3.Qd7+ Kb6 4.Qd8+ Kb5 5.Qb8+ Ka5 [not Kc5 6.Qd6 Kd4?? 7.Qxb4] 6.Qc7+ Rb6 7.Qc5+ Rb5 8.Qc7+ Kb4 and you go for the hills.
edit: 5.a4+!, noticed by @jknappen, is problematic. The right solution is rather to bring the king to a7 before covering checks with ...Rb7, as in @InertialIgnorance 's answer which should become the accepted one.
To escape from perpetual check when up material in an endgame, a good strategy is to try to surround your King with your extra pieces. In this case, that would be your Queen and Rook, and maybe a few of your pawns. Sample line:
1...Kb8 2. Qd6+ Ka7 3. Qc7+ Rb7 4. Qc5+ Kb8 and now White has a few options:
a) 5. Qd6+ Kc8 6. Qc6+ Rc7 (7. Qe8+ Kb7) 7. Qa8+ Kd7 and Black escapes.
b) 5. Qf8+ Kc7 6. Qe7+ (6. Qd6+ Kc8 transposes to line a) Kb6 7. Qd6+ Kb5!! 8. a4+ Ka5 9. Qc5+ Rb5!! and Black wins. White cannot take Rb5 with the a-pawn due to the threat of Qh1#
I found all the moves myself up until move 7 in line b). Komodo came up with the brilliant idea of 7...Kb5, 8...Ka5, and 9...Rb5