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Refined asymptotics for the blow-up of \(u_ t-{\Delta}u=u^ p\). (English) Zbl 0784.35010

Summary: This work is concerned with positive, blowing-up solutions of the semilinear heat equation \(u_ t-\Delta u=u^ p\) in \(\mathbb{R}^ n\). Our main contribution is a sort of center manifold analysis for the equation in similarity variables, leading to refined asymptotics for \(u\) in a backward space-time parabola near any blow-up point. We also explore a connection between the asymptotics of \(u\) and the local geometry of the blow-up set.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
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