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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 $$u_t=\Delta u+ f(u),\quad x \in \mathbb{R}^n,\ t>0 $$ with initial data $u_0$ satisfying $-N\leq u_0\leq M$, $u_0 \not\equiv M$ and $\lim_{|x|\to \infty} \inf_{x\in B_m} u_0(x)=M$, where $M+N>0$, $f(M)>0$ and radius of ball $B_m$ diverges to the infinity as $m\to \infty$. The nonlinear term $f$ is assumed to be Lipschitz in $\mathbb{R}$ and $\liminf_{s \to \infty}f(s)/s^p>0$ for some $p>1$, $f'\geq 0$. 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.

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)