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 liminf f(u)/u^p > 0 for some p > 1 is considered when initial data u_0 satifies u_0 leq M, u_0 e M and lim_{ightarrow infty} 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.

Keywords

• nonlinear heat equation
• blow-up at space infinity
• blow-up direction.