Marc Anthony and Jennifer Lopez
07/18/2011 AT 03:00 PM EDT
07/18/2011 AT 01:30 PM EDT
One of the last times they were in the public eye together, Jennifer Lopez
and Marc Anthony were sharing the stage – and a passionate kiss – at the May 25 American Idol
But after news broke Friday that the power couple were ending their marriage
, a source close to the family tells PEOPLE that Anthony, 42, and Lopez, 41, who have two children, 3-year-old twins Max and Emme, together, split "after months of nonstop arguing."
"Marc and Jennifer decided it's best to go their separate ways for the sake of their two kids," the source says.
Anthony spent the weekend in Colombia, where he put his marital woes aside to put on an upbeat performance
. The only reference to his marriage came in the form of a joke. "They're saying I'm single," he told the audience.
Meanwhile, Lopez was back to work Sunday at Milk Studios in L.A., where she was shooting an ad campaign for her Kohl's fashion line.
"Jennifer is doing okay," the source says. "In the past two weeks, she has spent a lot of quality time with her kids and her family in L.A."
The source adds: "She is, of course, sad that she and Marc didn't work out, but they both feel they have tried everything."