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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

${u}_{t}={\Delta }u+f\left(u\right),\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{n},\phantom{\rule{4pt}{0ex}}t>0$

with initial data ${u}_{0}$ satisfying $-N\le {u}_{0}\le M$, ${u}_{0}¬\equiv M$ and ${lim}_{|x|\to \infty }{inf}_{x\in {B}_{m}}{u}_{0}\left(x\right)=M$, where $M+N>0$, $f\left(M\right)>0$ and radius of ball ${B}_{m}$ diverges to the infinity as $m\to \infty$. The nonlinear term $f$ is assumed to be Lipschitz in $ℝ$ and ${lim inf}_{s\to \infty }f\left(s\right)/{s}^{p}>0$ for some $p>1$, ${f}^{\text{'}}\ge 0$. 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:
 35K55 Nonlinear parabolic equations 35K05 Heat equation 35K15 Second order parabolic equations, initial value problems 35B40 Asymptotic behavior of solutions of PDE 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
##### Keywords:
similinear heat equation; blow up; sub-super solution