Southern Railway has "strongly advised" passengers not to travel on Tuesday, Wednesday or Friday after losing a court bid to halt drivers' strikes.
The company lost their appeal to prevent the strike by members of the Aslef union.
Southern said due to the timing of the court's decision on Monday it will be too late to get trains and drivers in the correct position and any services it can run will be extremely limited.
Speaking before the Court of Appeal announced their decision, Angie Doll, passenger services director for Southern, said: "Even if are able to stop the strikes through the court, services will still be very heavily impacted tomorrow.
"We will work through the night to try and provide as many services as possible, but we are still advising passengers not to travel as we will not be able to offer a robust service they can rely on.
"We are sorry but the industrial action by Aslef leaves us no viable alternative, but be assured we will do all we can overnight."