## 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
Full Text: