Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in RN
Kiyoshi Mochizuki, Ryuichi Suzuki · Journal of the Mathematical Society of Japan · 1992
In this paper we shall consider the Cauchy problem (0.1)Laplacianand$\beta(v),$ $f(v)$ with $v\geqq 0$ and $u_{0}(x)$ are nonnegative functions.Equation (0.1) describes the combustion process in a stationary medium, in which the thermal conductivity $\beta'(u)^{-1}$ 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.Throughout this paper we assume (A1) $\beta(v),$ $f(v)\in C^{\infty}(R_{+})\cap C(\overline{R}_{+})$ , where $R_{+}=(0, \infty)$ and $\overline{R}_{+}=[0, \infty)$ ; $\beta(v)>0,$ $\beta'(v)>0,$ $\beta'(v)\leqq 0$ and $f(v)>0$ for $v>0; \lim_{varrow\infty}\beta(v)=\infty$ ; $f\circ\beta^{-1}$ is locally Lipschitz continuous in $[\beta(0),$ $\infty)$ .(A2) $u_{0}(x)\geqq 0,$ $ ot\equiv 0$ and $\in B(R^{N})$ (bounded continuous in $R^{N}$ ).With these conditions the above Cauchy problem has a unique local solution $u(x, t)$ (in time) which satisfies (0.1) in $R^{N}\cross(0, T)$ in the following weak sense (see $e.g.$ , Oleinik et al. [17]), where $T>0$ is assumed sufficiently small.T)$ , and $\in B(\overline{G}\cross[0, \tau])$ for each $0<\tau<T$ .2) For any bounded domain $\Omega\subset G,$ $0<\tau