Quintic NLW/NLKG on R3: Difference between revisions
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* Scaling is <math>s=1</math>. Thus this equation is energy-critical. | |||
* Scaling is <math>s=1</math>. | * LWP for <math>s \geq 1</math> by Strichartz estimates (see e.g. [[LbSo1995]]; earlier references exist) | ||
* LWP for <math>s \geq 1</math> by Strichartz estimates (see e.g. [[ | |||
** 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. | ||
** 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 [[ | * GWP for <math>s=1</math> in the defocussing case ([[Gl1990]], [[Gl1992]]). The main new ingredient is energy non-concentration ([[Sw1988]], [[Sw1992]]). | ||
** Further decay estimates and scattering were obtained in [[ | ** Further decay estimates and scattering were obtained in [[BaSa1998]]; global Lipschitz dependence was obtained in [[BaGd1997]]. | ||
** For smooth data GWP and scattering was shown in [[ | ** For smooth data GWP and scattering was shown in [[Gl1992]]; see also [[SaSw1994]] | ||
** For radial data GWP and scattering was shown in [[ | ** For radial data GWP and scattering was shown in [[Sw1988]] | ||
** For data with small energy this was shown for general quintic non-linearities (and for either NLW or NLKG) in [[ | ** For data with small energy this was shown for general quintic non-linearities (and for either NLW or NLKG) in [[Ra1981]]. | ||
** Global weak solutions can be constructed by general methods (e.g. [[ | ** Global weak solutions can be constructed by general methods (e.g. [[Sr1989]], [[Sw1992]]); uniqueness was shown in [[Kt1992]] | ||
** 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]]. | ||
** When there is a convex obstacle GWP for smooth data is known [[ | ** When there is a convex obstacle GWP for smooth data is known [[SmhSo1995]]. | ||
[[Category:Wave]] | |||
[[Category:Equations]] | |||
Revision as of 07:12, 2 August 2006
- Scaling is . Thus this equation is energy-critical.
- 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 in the defocussing case (Gl1990, Gl1992). The main new ingredient is energy non-concentration (Sw1988, Sw1992).
- Further decay estimates and scattering were obtained in BaSa1998; global Lipschitz dependence was obtained in BaGd1997.
- For smooth data GWP and scattering was shown in Gl1992; see also SaSw1994
- For radial data GWP and scattering was shown in Sw1988
- For data with small energy this was shown for general quintic non-linearities (and for either NLW or NLKG) in Ra1981.
- Global weak solutions can be constructed by general methods (e.g. Sr1989, Sw1992); uniqueness was shown in Kt1992
- In the focussing case there is blowup from large data by the ODE method.
- When there is a convex obstacle GWP for smooth data is known SmhSo1995.
