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Author Collot, Charles, 1990- author.

Title On the stability of type I blow up for the energy super critical heat equation / Charles Collot, Pierre Raphaël, Jeremie Szeftel
Published Providence, RI : American Mathematical Society, 2019
Online access available from:
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Description 1 online resource (v, 97 pages)
Series Memoirs of the American Mathematical Society, 1947-6221 ; volume 260, number 1255
Memoirs of the American Mathematical Society ; no. 1255
Contents Cover; Title page; Chapter 1. Introduction; 1.1. Setting of the problem; 1.2. Type I and type II blow up; 1.3. Statement of the result; Acknowledgments; Notations; Organization of the paper; Chapter 2. Construction of self-similar profiles; 2.1. Exterior solutions; 2.2. Constructing interior self-similar solutions; 2.3. The matching; Chapter 3. Spectral gap in weighted norms; 3.1. Decomposition in spherical harmonics; 3.2. Linear ODE analysis; 3.3. Perturbative spectral analysis; 3.4. Proof of Proposition 3.1; Chapter 4. Dynamical control of the flow; 4.1. Setting of the bootstrap
4.2. ̂{∞} bound4.3. Modulation equations; 4.4. Energy estimates with exponential weights; 4.5. Outer global ² bound; 4.6. Control of the critical norm; 4.7. Conclusion; 4.8. The Lipschitz dependence; Appendix A. Coercivity estimates; Appendix B. Proof of (4.43); Appendix C. Proof of Lemma 3.2; Appendix D. Proof of Lemma 3.3; Bibliography; Back Cover
Summary The authors consider the energy super critical semilinear heat equation \partial _{t}u=\Delta u+û{p}, x\in \mathbb{R}̂3, p>5. The authors first revisit the construction of radially symmetric self similar solutions performed through an ode approach and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. They then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional nonradial stability of these solutions for smooth well localized initial data usi
Bibliography Includes bibliographical references
Notes Online resource; title from PDF title page (viewed August 27, 2019)
Subject Heat equation.
Heat equation.
Partial differential equations -- Elliptic equations and systems -- Semilinear elliptic equations.
Partial differential equations -- Parabolic equations and systems -- Semilinear parabolic equations.
Partial differential equations -- Qualitative properties of solutions -- Bifurcation.
Partial differential equations -- Qualitative properties of solutions -- Blow-up.
Partial differential equations -- Qualitative properties of solutions -- Stability.
Form Electronic book
Author Raphaël, Pierre, 1975- author.
Szeftel, Jérémie, 1977- author.
ISBN 1470453347