Quintic NLW/NLKG on R2: Difference between revisions

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* Scaling is <math>s_c = 1/2</math>.
* Scaling is <math>s_c = 1/2</math>.
* LWP for <math>s \geq 1/2</math> by Strichartz estimates (see e.g. [[Bibliography#LbSo1995|LbSo1995]]; earlier references exist)
* LWP for <math>s \geq 1/2</math> by Strichartz estimates (see e.g. [[LbSo1995]]; earlier references exist)
** When <math>s=1/2</math> the time of existence depends on the profile of the data and not just on the norm.
** When <math>s=1/2</math> the time of existence depends on the profile of the data and not just on the norm.
** For <math>s<s_c</math> one has instantaneous blowup in the focusing case, and unbounded growth of <math>H^s</math> norms in the defocusing case [CtCoTa-p2]
** For <math>s<s_c</math> one has instantaneous blowup in the focusing case, and unbounded growth of <math>H^s</math> norms in the defocusing case ([[CtCoTa-p2]])
* GWP for <math>s > 3/4</math> for defocussing NLW/NLKG [Fo-p]
* GWP for <math>s > 3/4</math> for defocussing NLW/NLKG ([[Fo-p]])
** For <math>s \geq 1</math> this follows energy conservation.
** For <math>s \geq 1</math> this follows energy conservation.
** One also has GWP and scattering for data with small <math>H^{1/2}</math> norm for general quintic non-linearities (and for either NLW or NLKG).
** One also has GWP and scattering for data with small <math>H^{1/2}</math> norm for general quintic non-linearities (and for either NLW or NLKG).
** In the focussing case there is blowup from large data by the ODE method.
** In the focussing case there is blowup from large data by the ODE method.


[[Category:Wave]]
----  [[Category:Equations]]
[[Category:Equations]]

Latest revision as of 07:10, 2 August 2006

  • Scaling is .
  • LWP for by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
    • When the time of existence depends on the profile of the data and not just on the norm.
    • For one has instantaneous blowup in the focusing case, and unbounded growth of norms in the defocusing case (CtCoTa-p2)
  • GWP for for defocussing NLW/NLKG (Fo-p)
    • For this follows energy conservation.
    • One also has GWP and scattering for data with small norm for general quintic non-linearities (and for either NLW or NLKG).
    • In the focussing case there is blowup from large data by the ODE method.