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Zbl 0805.35065
Mochizuki, Kiyoshi; Suzuki, Ryuichi
Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in $\bbfR\sp N$. (English)
[J] J. Math. Soc. Japan 44, No.3, 485-504 (1992). ISSN 0025-5645; ISSN 1881-1167; ISSN 1881-2333/e

From the introduction: We consider the Cauchy problem $$\partial\sb t \beta (u) = \Delta u + f(u)\quad \text { in } (x,t) \in \bbfR\sp N \times (0,T), \quad u(x,0) = u\sb 0(x)\quad \text { in } x \in \bbfR\sp N, \tag 1 $$ where $\partial\sb t = \partial/ \partial t$, $\Delta$ is the $N$- dimensional Laplacian and $\beta (v)$, $f(v)$ with $v \ge 0$ and $u\sb 0(x)$ are nonnegative functions. Equation (1) describes the combustion process in a stationary medium, in which the thermal conductivity $\beta' (u)\sp{-1}$ and the volume heat source $f(u)$ are depending in a nonlinear way on the temperature $\beta (u) = \beta( u(x,t))$ 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 [{\it 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 2000:: *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

Citations: Zbl 0789.35024

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