An RSVP LSP can deal with a failure in two ways:
If the LSP is a loosely routed LSP, then RSVP finds a new IGP path around the failure, and traffic follows this new path. This may involve some churn in the network if the LSP comes down and then gets re-routed. The tunnel damping feature was implemented on the LSP so that all the dependent protocols and applications do not flap unnecessarily.
If the LSP is a CSPF-computed LSP with the fast reroute option enabled, then RSVP switches to the detour path very quickly. From that point, a new LSP is attempted from the head-end (global revertive). When the new LSP is in place, the traffic switches over to the new LSP with make-before-break.