Nearly two years after the oil rig explosion that killed 11 people and spilled millions of barrels of crude oil into the Gulf of Mexico, the myriad plaintiffs suing BP and other companies over the disaster are about to get their day in court.

With the start of the high-profile trial set for next Monday, and the specter of potential liability that some experts have estimated at $40 billion, BP and other defendants are stepping up negotiations to end the litigation before Judge Carl J. Barbier of Federal District Court picks up his gavel.