Liu, Wenxiong The blow-up rate of solutions of semilinear heat equations. (English) Zbl 0672.35035 J. Differ. Equations 77, No. 1, 104-122 (1989). 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 Y. Giga and 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)]. Reviewer: C.Popa Cited in 26 Documents MSC: 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations Keywords:asymptotic behavior; semilinear; blow up; Cauchy problem; energy estimate; Pohozaev’s type identity Citations:Zbl 0585.35051; Zbl 0601.35052 PDFBibTeX XMLCite \textit{W. Liu}, J. Differ. Equations 77, No. 1, 104--122 (1989; Zbl 0672.35035) Full Text: DOI References: [2] Bebernes, J.; Kassoy, D., A mathematical analysis of blow-up for thermal reactions, SIAM J. Appl. Math., 40, 476-484 (1981) · Zbl 0481.35048 [6] Friedman, A.; Mcleod, B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-447 (1985) · Zbl 0576.35068 [7] Giga, Y.; Kohn, R. V., Asymptotic self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38, 297-319 (1985) · Zbl 0585.35051 [8] Giga, Y.; Kohn, R. V., Characterizing blow-up using similarity variables, Indiana Univ. Math. J., 36, 1-44 (1987) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.