On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation
Seiten
2019
American Mathematical Society (Verlag)
978-1-4704-3626-1 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-3626-1 (ISBN)
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The authors consider the energy super critical semilinear heat equation $/partial _{t}u=/Delta u u^{p}, x/in /mathbb{R}^3, p>5.$. The authors draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.
The authors consider the energy super critical semilinear heat equation $/partial _{t}u=/Delta u u^{p}, x/in /mathbb{R}^3, p>5.$ 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 using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.
The authors consider the energy super critical semilinear heat equation $/partial _{t}u=/Delta u u^{p}, x/in /mathbb{R}^3, p>5.$ 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 using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.
Charles Collot, Universite de Nice-Sophia Antipolis, France. Pierre Raphael, Universite de Nice-Sophia Antipolis, France. Jeremie Szeftel, Universite Paris 6, France.
Introduction
Construction of self-similar profiles
Spectral gap in weighted norms
Dynamical control of the flow
Appendix A. Coercivity estimates
Appendix B. Proof of (6.7)
Appendix C. Proof of Lemma 2.1
Appendix D. Proof of Lemma 2.2
Bibliography.
| Erscheinungsdatum | 02.09.2019 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 209 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 1-4704-3626-4 / 1470436264 |
| ISBN-13 | 978-1-4704-3626-1 / 9781470436261 |
| Zustand | Neuware |
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