Giga, Yoshikazu; Kohn, Robert V. Characterizing blowup using similarity variables. (English) Zbl 0601.35052 Indiana Univ. Math. J. 36, 1-40 (1987). We bound the growth rate and characterize the asymptotic behavior at blow up of solutions of \(u_ t-\Delta u-f(u)=0\), when \(f(u)\sim | u| ^{p-1}u\) as \(| u| \to \infty\). The analysis uses energy-type identities for a rescaled equation, obtained from the original one by introducing similarity variables. As an application we prove a new lower bound on the blow up rates of certain norms of u. All results are restricted to subcritical \(p: 1<p<(n+2)/(n-2)\) or \(n\leq 2\), where n is the space dimension. Cited in 4 ReviewsCited in 191 Documents MSC: 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35A30 Geometric theory, characteristics, transformations in context of PDEs Keywords:asymptotic behavior; blow up; similarity variables; semilinear heat equations PDFBibTeX XMLCite \textit{Y. Giga} and \textit{R. V. Kohn}, Indiana Univ. Math. J. 36, 1--40 (1987; Zbl 0601.35052) Full Text: DOI