A B737-300 PAX CAUSED A LAVATORY FIRE BY SMOKING INFLT.

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PAX SMOKING IN AFT LAVATORY INFLT AND CAUSING FIRE. AFTER LNDG IN ZZZ; I WROTE UP THE USE OF A FIRE EXTINGUISHER BOTTLE; USED FOR AN INFLT FIRE IN THE AFT LAVATORY TRASH BIN AND SAID IT WAS OK; CALLED MAINT AND ADVISED THEM. AT NEXT STOP; DISPATCH ASKED THAT I CALL THEM; WHICH I DID; AND ADVISED THEM WHAT HAPPENED. I GUESS DISPATCH DIDN'T ADVISE PROPER AUTH OF INFLT FIRE. I SHOULD HAVE CALLED THEM AT ZZZ; BUT THOUGHT MAINT WOULD. IN FUTURE; I'LL CALL AND ADVISE DISPATCH TO ENSURE PROPER RPTING. 1/2 WAY TO ZZZ ONE OF THE FLT ATTENDANTS CALLED ME ON THE INTERPHONE AND SAID THERE WAS A FIRE IN THE AFT LAVATORY TRASH BIN AND THEY WERE PUTTING IT OUT AND THAT THEY WOULD KEEP ME INFORMED. LATER THEY CALLED AND SAID IT WAS PUT OUT OK AND THAT A MALE PAX CAUSED IT AND SAID HE ADMITTED SMOKING IN THE LAVATORY; BUT DID NOT PUT IT IN THE TRASH BIN. HE GAVE HIS NAME AND WE REQUESTED A CUSTOMER SVC SUPVR AND SECURITY TO MEET THE ACFT IN ZZZ. HE WAS GREETED BY THEM; TAKEN OFF (HE WAS GETTING OFF IN ZZZ ANYWAY; I BELIEVE). 4 EXTINGUISHER BOTTLES REPLACED; AFT LAVATORY INSPECTOR BY MAINT; LOGBOOK ANNOTATION MADE AND SIGNED OFF AND DEPARTED UNEVENTFULLY. OUR 4TH FLT ATTENDANT JUMP SEATER SAID LATER HE WAS 'ARRESTED' IN ZZZ1 FOR SMOKING IN A RESTROOM AT THE ARPT; SO IT MAY BE WORTH CHKING OUT WITH THE ZZZ1 ARPT POLICE.