Blow-up problems for a semilinear heat equation with large diffusion
The paper deals with the Cauchy-Neumann problem \[ \partial_t u = D \Delta u + u^p \quad \text{ for } x \in \Omega, \; 0<t<T_D, \] \[ \partial_{\nu} u(x,t) = 0 \quad \text{ for } x \in \partial \Omega, \; 0<t<T_D, \qquad u(x,0) = \phi(x) \quad \text{ for } x \in \Omega, \] where \(p>1\), \(\Omega\) is a bounded smooth domain in \(\mathbb R^N\) and \(\nu\) is the exterior unit normal to \(\partial \Omega\). Denoting by \(u_D(x,t)\) the solution corresponding to the fixed diffusive constant \(D>0\), the authors define the blow up time \(T_D\) as \[ T_D = \sup_{\tau >0}\{ u_D \in L^{\infty} ( \Omega \times (0,\tau))\} \] and the blow up set \(B_D\) as \[ \begin{multlined} B_D = \{ x \in \bar \Omega: \lim_{k \to \infty} (x_k, t_k) = (x, T_D), \\ \lim_{k \to \infty} u_D (x_k, t_k) = \infty \text{ for some sequence } (x_k, t_k) \in \bar \Omega \times (0,T_D) \}.\end{multlined} \] The function \[ u^*_D (x) = \lim_{t \nearrow T_D} u_D(x,t) \quad \text{for } x \in \bar \Omega \setminus B_D \] is called the blow up profile for the solution \(u_D\). The aim of the paper is to study the blow up time \(T_D\), the location of the blow up set \(B_D\) and the blow up profile \(u^*_D\) for sufficiently large \(D\). The paper is related to the earlier papers [\textit{H. Fujita}, J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109--124 (1966; Zbl 0163.34002)] and [\textit{K. Ishige}, Adv. Differ. Equ. 7, No. 8, 1003--1024 (2002; Zbl 1036.35096)] quoted by the authors.
scientific article; zbMATH DE number 2164825
scientific article