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Blow-up directions for quasilinear parabolic equations. (English) Zbl 1167.35393
Summary: We consider the Cauchy problem for quasilinear parabolic equations ${u}_{t}={\Delta }\varphi \left(u\right)+f\left(u\right)$, with the bounded non-negative initial data ${u}_{0}\left(x\right)\left({u}_{0}\left(x\right)¬\equiv 0\right)$, where $f\left(\xi \right)$ is a positive function in $\xi >0$ satisfying a blow-up condition ${\int }_{1}^{\infty }1/f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}d\xi <\infty$. 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$v/\text{d}t=f\left(v\right)$ with the initial data $\parallel {u}_{0}{\parallel }_{{L}^{\infty }\left({ℝ}^{N}\right)}>0$. 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 $f\left(\xi \right)$ grows more rapidly than $\varphi \left(\xi \right)$.
##### MSC:
 35K55 Nonlinear parabolic equations 35K15 Second order parabolic equations, initial value problems 35K65 Parabolic equations of degenerate type 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
##### Keywords:
non-negative solutions; least blow-up time