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Zbl 1173.35531
Giga, Yoshikazu; Umeda, Noriaki
Blow-up directions at space infinity for solutions of semilinear heat equations.
(English)
[J] Bol. Soc. Parana. Mat. (3) 23, No. 1-2, 9-28 (2005); correction ibid. 24, No. 1-2, 19-24 (2006). ISSN 0037-8712

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.
[Yavdat Ilyasov (Ufa)]
MSC 2000:
*35K55 Nonlinear parabolic equations
35K05 Heat equation
35K15 Second order parabolic equations, initial value problems
35B40 Asymptotic behavior of solutions of PDE
35B05 General behavior of solutions of PDE

Keywords: similinear heat equation; blow up; sub-super solution

Cited in: Zbl 1185.35116

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