*(English)*Zbl 0805.35065

From the introduction: We consider the Cauchy problem

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 $\beta \left(u\right)=\beta \left(u\right(x,t\left)\right)$ 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.