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Characterizing blowup using similarity variables. (English) Zbl 0601.35052

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.

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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