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Zbl 1167.35393
Seki, Yukihiro; Umeda, Noriaki; Suzuki, Ryuichi
Blow-up directions for quasilinear parabolic equations. (English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 138, No. 2, 379-405 (2008). ISSN 0308-2105; ISSN 1473-7124/e

Summary: We consider the Cauchy problem for quasilinear parabolic equations $u_t=\Delta\varphi(u)+f(u)$, with the bounded non-negative initial data $u_0(x) (u_0(x)\not\equiv 0)$, where $f(\xi)$ is a positive function in $\xi>0$ satisfying a blow-up condition $\int_1^{\infty}1/f(\xi)\,d\xi<\infty$. We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation d$v/\text {d} t=f(v)$ with the initial data $\Vert u_0\Vert _{L^{\infty}(\Bbb{R}^N)}>0$. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on $u_0$ for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on $u_0$ for blow-up with the least blow-up time, provided that $f(\xi)$ grows more rapidly than $\varphi(\xi)$.

MSC 2000:: *35K55 Nonlinear parabolic equations
35K15 Second order parabolic equations, initial value problems
35K65 Parabolic equations of degenerate type
35B05 General behavior of solutions of PDE

Keywords: non-negative solutions; least blow-up time

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Zentralblatt MATH Berlin [Germany]