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On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation
 
Charles Collot Université de Nice-Sophia Antipolis, Nice, France
Pierre Raphaël Université de Nice-Sophia Antipolis, Nice, France
Jérémie Szeftel Université Paris 6, France
On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation
Softcover ISBN:  978-1-4704-3626-1
Product Code:  MEMO/260/1255
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5334-3
Product Code:  MEMO/260/1255.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3626-1
eBook: ISBN:  978-1-4704-5334-3
Product Code:  MEMO/260/1255.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation
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On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation
Charles Collot Université de Nice-Sophia Antipolis, Nice, France
Pierre Raphaël Université de Nice-Sophia Antipolis, Nice, France
Jérémie Szeftel Université Paris 6, France
Softcover ISBN:  978-1-4704-3626-1
Product Code:  MEMO/260/1255
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5334-3
Product Code:  MEMO/260/1255.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3626-1
eBook ISBN:  978-1-4704-5334-3
Product Code:  MEMO/260/1255.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2602019; 93 pp
    MSC: Primary 35

    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.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Construction of self-similar profiles
    • 3. Spectral gap in weighted norms
    • 4. Dynamical control of the flow
    • A. Coercivity estimates
    • B. Proof of (4.43)
    • C. Proof of Lemma
    • D. Proof of Lemma
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2602019; 93 pp
MSC: Primary 35

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.

  • Chapters
  • 1. Introduction
  • 2. Construction of self-similar profiles
  • 3. Spectral gap in weighted norms
  • 4. Dynamical control of the flow
  • A. Coercivity estimates
  • B. Proof of (4.43)
  • C. Proof of Lemma
  • D. Proof of Lemma
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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