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Bubbling Blow-Up in Critical Parabolic Problems

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Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2186))

Abstract

These lecture notes are devoted to the analysis of blow-up of solutions for some parabolic equations that involve bubbling phenomena. The term bubbling refers to the presence of families of solutions which at main order look like scalings of a single stationary solution which in the limit become singular but at the same time have an approximately constant energy level. This arise in various problems where critical loss of compactness for the underlying energy appears. Three main equations are studied, namely: the Sobolev critical semilinear heat equation in \(\mathbb{R}^{n}\), the harmonic map flow from \(\mathbb{R}^{2}\) into S 2, the Patlak-Keller-Segel system in \(\mathbb{R}^{2}\).

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Acknowledgements

This work has been supported by Fondecyt grant 115066, Millennium Nucleus Center for Analysis of PDE NC13001, and Fondo Basal CMM.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Manuel del Pino

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  1. Manuel del Pino
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Correspondence to Manuel del Pino .

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  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

    Matteo Bonforte

  2. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Gabriele Grillo

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del Pino, M. (2017). Bubbling Blow-Up in Critical Parabolic Problems. In: Bonforte, M., Grillo, G. (eds) Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Lecture Notes in Mathematics(), vol 2186. Springer, Cham. https://doi.org/10.1007/978-3-319-61494-6_2

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