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)