Blow-up directions at space infinity for solutions of semilinear heat equations

A blowing up solution of the semilinear heat equation ut = Delta u+ f (u) with f satisfying lim inf f (u)/u(p) > 0 for some p > 1 is considered when initial data u(0) satisfies u(0) <= M, u(0) not equivalent to M and lim(m) (>infinity) inf(x)is an element of B-m u(0)(x) = M with sequence of ball {B-m} whose radius diverging to infinity. It is shown that the solution blows up only at space infinity. A notion of blow-up direction is introduced. A characterization for blow-up direction is also established.