The blow-up rate of solutions of semilinear heat equations
This paper deals with the asymptotic behavior of the solution of the semilinear equation \[ u_ t-\Delta u=e^ u\quad in\quad \Omega \times (0,T),\quad u=-K\leq 0\quad on\quad \partial \Omega \times (0,T), \] \[ u(x,0)=\phi (x)\geq -K\quad for\quad x\in \Omega \subset {\mathbb{R}}^ n, \] near the blow up points as \(t\to T\). The main result (Theorem 5.4) states that this behavior is of the logarithmic type. More precisely, let \(\Omega\) be a convex domain and \(\Delta \phi +e^{\phi}\geq 0.\) If \(n\leq 2\) and a is a blow-up point (i.e. there exists a sequence \(\{(x_ m,t_ m)\}\) such that \(x_ m\to a\), \(t_ m\to T\) and \(u(x_ m,t_ m)\to +\infty)\), then u(x,t)-log(1/(T-t))\(\to 0\) as \(t\to T\), uniformly for \(| x-a| \leq C(T-t)^{1/2},\) where C is any positive constant. The author also proves (for any dimension n) the nondegeneracy of any blow-up point. In the case of the Cauchy problem \((\Omega ={\mathbb{R}}^ n)\), the blow-up set is bounded, provided \(\phi\) satisfies some growth restrictions at \(\infty\). In order to prove the main result, an energy estimate and a Pohozaev's type identity are also established. Note that analogous results for the semilinear heat equation \(u_ t-\Delta u=| u|^{p-1}u\) was obtained by \textit{Y. Giga} and \textit{R. V. Kohn} [Commun. Pure Appl. Math. 38, 297-319 (1985; Zbl 0585.35051) and Indiana Univ. Math. J. 36, 1-40 (1987; Zbl 0601.35052)].
scientific article; zbMATH DE number 4099791
scientific article