B747 Captain reported a partial loss of power on the #4 engine during climb. After communicating with dispatch and maintenance; the crew decided to shutdown the engine and proceed to their destination where appropriate maintenance action could be completed.

Narrative

This report is from the Captain's point of view:ZZZZ - ZZZMax takeoff weightNo Hazmat on boardFO A was the PF (Pilot Flying)CA was PM (Pimlot Monitoring)FO B was on third observer seat While climbing out of ZZZZ; ATC cleared us to climb FL310. T0-1 Climb; ECON speed; not exceeding any limitation nor doing anything out of the normal SOP operation; experience a partial power loss on Engine #4. FO A was the PF; and I directed him to keep the controls.I transferred the radios to FO B (Observer) and directed him to watch FO A (PF) while I was heads down trying to sort out what could have happened. We requested to level off at FL270 while we run appropriate checklists and run a diagnostics of the issue we were facing. I (CA/PM) noticed that the N1 indication had rolled back to about 55% while every other parameters (fuel flow/oil pressure/oil temp) met the given N1 rotation; except the EGT that fluctuated between 300-400-500C. At first to me it seemed like a fuel control issue; but not one EICAS message showed up. So up to the point we were uncertain of what had happened. We don't remember hitting anything on the climb. Decided to call dispatch to keep them updated of what had just happened.MX (Maintenance) Control got on the call; I explained the scenario to them; they asked me a few questions as far as what the parameters were; I gave all the information they needed. MX Control said they were checking on their end; to see if they had some live information on their fancy engine monitoring program. I asked them if we could go ahead and shut down the engine; which at the moment they said not to shut down.FO B and A coordinated with ZZZZ1 [Radar Control] to stay inside their FIR; just in case we needed to return.While MX Control was running their diagnostics; we got a Status Message : Eng 4 Fire Loop A (first)I passed that Status Message to dispatch and Mx Control.Shortly after the EGT #4 indications blanked out. I advised MX Control that in case we lost the second Fire Detection System I was going to exercise my Captain's authority and shutdown the said engine. Our thought process at the time was: we didn't want to have an engine potentially catch on fire without any EICAS indication; as we had a lot of fuel; and dumping fuel while an engine was on fire is not a very smart idea.While we were waiting for an answer from the people at Maintenance Control; whether or not to shut down the engine/divert and what not; what we feared came true. A second status message: Eng 4 Fire Loop B.I advised MX Control we would be shutting down the engine; and so we did. As a crew we discussed that returning to ZZZZ wasn't the best course of action; as we had so much fuel to get rid of to land safely below max landing weight. We agreed that proceeding to ZZZZ2 was the best option; as ZZZZ2 has 3 long runways and a state of the art maintenance control (and a little bit because broken airplanes in ZZZZ historically cause a lot of trouble). In case we had to land in ZZZZ and block off one of their runways they wouldn't be happy. But our decision was mainly based on safely getting the airplane on the ground causing the least amount of trouble possible. With the information we had at the time; we couldn't figure out what was going on; because it still had N1 rotation; no EICAS messages or anything. So sincerely; we were under the impression that it was a partial loss of power; maybe related to fuel feeding. Many things went through my head; more things that I could speak to my crew about at the time. Initially; I even thought that a cable on the throttle quadrant could have snapped (It happened to me when I flew a piston); but we have throttle authority up to 50-55% of N1. We tried it all; but not once did we play heroes and tried to push to complete the mission".It was a joint decision between the crew; dispatch; and MX Control that proceeding to ZZZZ2 was the plan we were going to execute; dispatch said he would run numbers for ZZZZ2; update the performance and send us a route. At that time; we ended the call. I took the radios back from FO B. Advised ZZZZ [Radar Control] that we would initially proceed as filed and divert to ZZZZ2 eventually; staying at FL270. They then directed us to fly an assigned heading. Transferred to ZZZZ3 [Radar Control] and advised them of the issue we had with an engine shutdown.Continuing the flight; ATC told us we could fly as fast as we wanted to get to our destination; but I truly didn't want to put the airplane through that unnecessary condition as we were safe; and the engine was already secured. I advised dispatch that we would need to dump fuel; and that we would do that with ZZZZ2. Transponder was switched to TA Only just in case we ran into a situation; as we were lower than normal for that route.Enroute to ZZZZ2; I read over the Severe Damage checklist; even though we didn't have an indication of severe damage. I elected not to pull the fire handle as directed by the checklist as the engine had already been safely shut down. I was later told by the mechanic in ZZZZ2 that the decision was a good idea; not sure why; but that's what he said in our debrief. About 1 hour from landing; we were given a fix; radial; distance; and altitude (FL180) to burn/dump the fuel. It took us approximately 37 minutes to dump/burn nearly 50 tons of fuel. After that; when we were predicting to land with about at max landing weight; we exited the hold and started getting vectors to the airport. Got a fuel imbalance 1-4 EICAS message; somewhat expected as the flow from reserve 2-3 into Main 2-3 is probably not as fast as the fuel jettison valves. We took care of the imbalance enroute to the airport by the time we got cleared for the approach; and just landed with 2.4 more on mains 2-3 than tanks 1-4. EICAS message Fuel Imbalance didn't go away; but we discussed; and decided that we didn't want to waste the time; nor request to hold to just balance that difference. We did speak of the potential risk of a go-around; but since it was CAVOK and winds were calm; we didn't anticipate a potential of a Go-Around. Safely landed; taxied to the parking spot.No assistance was needed after landing. After landing; ZZZZ2 MX came to the airplane to inspect the engine. It looked like the blades on the tail cone section of the engine were damaged. I could be wrong; but since the front fan blades were intact; the idea of FOD injection was out of the table. I'm not an expert; but I would blame that on metal fatigue.After the cowling was opened; pieces of some kind of ducting fell off from the engine; as if they were just loose inside the cowling. Even the mechanic couldn't believe that such a thing had happened. We wrote up in the logbook the 'Status Messages' we had; the things we had experienced on the flight.Big players:CRM. Had it not been for a really well trained flight crew; the outcome would have been different.We took everything in a safe and timely manner. Everyone did as it was expected of them; and performed at a really high standard. FO A and B. were very professional and dealt with this issue extremely calmly; that made a huge difference.Mx Control. Although Mx Control couldn't find anything on their end as far as engine parameters go; we jointly agree that shutting down the engine was the smart thing to do.Dispatch. Played a big role in this; they were on the spot with everything we needed.Deadheaders. We had crew members deadheading; they also showed up in the cockpit to offer help; which was much appreciated."