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Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in ${ℝ}^{N}$. (English) Zbl 0805.35065

From the introduction: We consider the Cauchy problem

${\partial }_{t}\beta \left(u\right)={\Delta }u+f\left(u\right)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\left(x,t\right)\in {ℝ}^{N}×\left(0,T\right),\phantom{\rule{1.em}{0ex}}u\left(x,0\right)={u}_{0}\left(x\right)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}x\in {ℝ}^{N},\phantom{\rule{2.em}{0ex}}\left(1\right)$

where ${\partial }_{t}=\partial /\partial t$, ${\Delta }$ is the $N$- dimensional Laplacian and $\beta \left(v\right)$, $f\left(v\right)$ with $v\ge 0$ and ${u}_{0}\left(x\right)$ are nonnegative functions. Equation (1) describes the combustion process in a stationary medium, in which the thermal conductivity ${\beta }^{\text{'}}{\left(u\right)}^{-1}$ and the volume heat source $f\left(u\right)$ are depending in a nonlinear way on the temperature $\beta \left(u\right)=\beta \left(u\left(x,t\right)\right)$ of the medium. The main purpose of the present paper is the study of blow-up solutions near the blow-up time. Especially, we are interested in the shape of the blow-up set which locates the “hot-spots” at the blow-up time. In addition, since our quasilinear equation (1) has a property of finite propagation, there are some interesting subjects such as the regularity of the interface and its asymptotic behavior near the blow-up time. These problems have been studied by one of the authors [R. Suzuki, Publ. Res. Inst. Math. Sci. 27, No. 3, 375-398 (1991; Zbl 0789.35024)], in the case $N=1$. This paper extends some of his results to higher dimensional problems.

##### MSC:
 35K65 Parabolic equations of degenerate type 35B40 Asymptotic behavior of solutions of PDE 80A25 Combustion, interior ballistics
##### Keywords:
blow-up solutions; blow-up time; blow-up set