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Blow-up directions at space infinity for solutions of semilinear heat equations. (English) Zbl 1173.35531

The paper deals with a blowing up solution of the semilinear heat equation

u t =Δu+f(u),x n ,t>0

with initial data u 0 satisfying -Nu 0 M, u 0 ¬M and lim |x| inf xB m u 0 (x)=M, where M+N>0, f(M)>0 and radius of ball B m diverges to the infinity as m. The nonlinear term f is assumed to be Lipschitz in and lim inf s f(s)/s p >0 for some p>1, f ' 0. In the main result authors show that the solution blows up only at the space infinity. Furthermore, authors introduce a notion of blow up direction at the space infinity and establish characterizations for blow up directions by profile of initial data.


MSC:
35K55Nonlinear parabolic equations
35K05Heat equation
35K15Second order parabolic equations, initial value problems
35B40Asymptotic behavior of solutions of PDE
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)