## 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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### References:

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