ACR WDB IN CRUISE FL310 GOT APU FIRE WARNING. EMER PROC FOLLOWED AND A DIVERSION TO MSP BEGUN. FUEL DUMPED TO GET DOWN TO LNDG WT. MAINT ADVISED HAD BEEN A PROB ON THIS ACFT. DEADHEADING CAPT WAS A GREAT ASSIST HANDLING COM WITH PAX. 2 MAN CREW VERY BUSY COMPLETING CHKLISTS AND FLYING THE ACFT.

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APPROX 40 MINS AFTER TKOF FROM DTW IN CRUISE; APU FIRE WARNING SOUNDED WITH ALL WARNING INDICATIONS. APU FIRE EMER PROC WAS EXECUTED AND THE FIRE BOTTLE DISCHARGED. FIRE WARNING REMAINED ON SO DIVERSION TO MSP WAS BEGUN. GROSS WT WAS APPROX 805000 LBS AND FUEL DUMP WAS STARTED AT THE SAME TIME APPROX XA05Z. MSP DISPATCH/MAINT CTL WAS CONTACTED AT THIS TIME AND MAINT CTL SAID AIRPLANE HAD HISTORY (PAST MONTH) OF 2 FALSE APU FIRE WARNINGS; 1 INFLT; 1 ON THE GND. WE DECLARED AN EMER AND HAD FIRE TRUCKS STANDING BY TO EXAMINE THE APU AREA UPON CLRING THE RWY. OVERWT LNDG WAS PLANNED; HOWEVER; USING MAX FUEL CRUISE AND DUMPING WE GOT WITHIN 1700 LBS OF MAX LNDG GROSS WT. LANDED MSP AT 631700 LBS RWY 11R; APPROX XA45Z. OFF DUTY CAPT ASSISTED GREATLY IN HANDLING CABIN COMS; PA'S; AND ALSO SOME COMPANY COM. FO WAS PF; CAPT PNF; UNTIL APCH CHK AND DECISION WAS MADE THAT CAPT WOULD ASSUME FLYING DUTIES. WHOLE INCIDENT WAS A STUDY IN GOOD CREW COORD AND POINTED TO THE FACT THAT 2 PLT CREWS GET VERY BUSY WHEN INCIDENTS OCCUR; EVEN IN GOOD WX. HELP OF THIRD CREW MEMBER GREATLY EVENED OUT THE LOAD. UPON LNDG; NO EVIDENCE OF FIRE OR SMOKE WAS FOUND BY THE FIRE TRUCKS SO ACFT WAS TAXIED TO THE GATE. ALL BEFORE TKOF; TKOF CLB; DSCNT; APCH; LNDG CHKS WERE PERFORMED STRICTLY BY SOP. APU HAD BEEN SHUT DOWN NORMALLY AFTER ENG START ON THE BEFORE TAXI CHKLIST. LNDG BEING OVER MAX GROSS WT WAS VERY SMOOTH AND FLAT AND NORMAL ROLLOUT ACCOMPLISHED.