Cubic NLW/NLKG on R4: Difference between revisions

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* Scaling is <math>s_c = 1</math>.
* Scaling is <math>s_c = 1</math>.
* LWP for <math>s \geq 1</math> by Strichartz estimates (see e.g. [[Bibliography#LbSo1995|LbSo1995]]; earlier references exist)
* LWP for <math>s \geq 1</math> by [[Strichartz estimate]]s (see e.g. [[LbSo1995]]; earlier references exist)
** When <math>s=1</math> the time of existence depends on the profile of the data and not just on the norm.
** When <math>s=1</math> the time of existence depends on the profile of the data and not just on the norm.
** One has strong uniqueness in the energy class [Pl-p5], [[Bibliography#FurPlTer2001|FurPlTer2001]]. This argument extends to other energy-critical and sub-critical powers in dimensions 4 and higher.
** One has strong uniqueness in the energy class [[Pl-p5]], [[FurPlTer2001]]. This argument extends to other energy-critical and sub-critical powers in dimensions 4 and higher.
** 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=1</math> in the defocussing case [[Bibliography#SaSw1994|SaSw1994]] (see also [[Bibliography#Gl1990|Gl1990]], [[Bibliography#Gl1992|Gl1992]], [[Bibliography#Sw1988|Sw1988]], [[Bibliography#Sw1992|Sw1992]], [[Bibliography#BaSa1998|BaSa1998]], [[Bibliography#BaGd1997|BaGd1997]]).
* GWP for <math>s=1</math> in the defocussing case [[SaSw1994]] (see also [[Gl1990]], [[Gl1992]], [[Sw1988]], [[Sw1992]], [[BaSa1998]], [[BaGd1997]]).
** 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]]

Revision as of 04:57, 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.
    • One has strong uniqueness in the energy class Pl-p5, FurPlTer2001. This argument extends to other energy-critical and sub-critical powers in dimensions 4 and higher.
    • For one has instantaneous blowup in the focusing case, and unbounded growth of norms in the defocusing case (CtCoTa-p2).
  • GWP for in the defocussing case SaSw1994 (see also Gl1990, Gl1992, Sw1988, Sw1992, BaSa1998, BaGd1997).
    • In the focussing case there is blowup from large data by the ODE method.
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