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

Abstract

A blowing up solution of the semilinear heat equation u_t = \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 \leq M, u\ne M and lim_{m\rightarrow infity} inf_{x\in 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.