life span of solutions for a semilinear heat equation with initial data having positive limit inferior at infinity

we present a new upper bound of the life span of positive solutions of a semilinear heat equation for initial data having positive limit inferior at space infinity. the upper bound is expressed by the data in limit inferior, not in every direction, but around a specific direction. it is also shown that the minimal time blow-up occurs when initial data attains its maximum at space infinity.