這個工作致力於建立時空的控制，對於一維時間五維空間臨界能量的非線性波方程
□u=u^(3/7)，

然後藉由使用質量和能量集中的技巧與來自[21]比較區間的估計來構築精確的Strichartz 控制

||u||_(L_t^(58/27) L_x^(29/6) (R^(1+5)))≤C(1+CE)^CE^100。

並且我們重新考慮在一維時間四維空間臨界能量的非線性波方程

□u=u^(3)，

根據能量在大球外面的使用技巧和來自區間大小比較的估計得到精確的Strichartz 控制

||u||_(L_t^(5/2) L_x^(20/3) (R^(1+4)))≤C(1+CE)^CE^24。

The work is devoted to establish the spacetime bound for the energy-critical nonlinear wave equation

□u=u^(3/7)

in R^(1+5).Then we build the explicit Strichartz bound

||u||_(L_t^(58/27) L_x^(29/6) (R^(1+5)))≤C(1+CE)^CE^100

for some absolute C>0 by using the techniques of the mass and energy concentration and interval-comparing estimate from [21].
And we reconsider the energy-critical nonlinear wave equation
□u=u^(3)

in R^(1+4).Thus we also build the explicit Strichartz bound

||u||_(L_t^(5/2) L_x^(20/3) (R^(1+4)))≤C(1+CE)^CE^24

for some absolute C>0 by using the techniques of the energy outside tha big ball and the method of the interval-size comparing from [21].

[1] H. Bahouri and P. G'erard. High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math., (19):131{175, 1999.
[2] J. Bourgain. Global well-posedness of defocusing 3D critical NLS in the radial case. JAMS, (12):145{171, 1999.
[3] M. Grillakis. Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. of Math., (132), 1990.
[4] M. Grillakis. Regularity for the wave equation with a critical nonlinearity. Commun. Pure Appl. Math., (45):749{774, 1992.
[5] M. Grillakis. A priori estimates and regularity of nonlinear waves. Proceed. Inter. Congress of Math., 1994.
[6] G. Sta lani H. Takaoka J. Colliander, M. Keel and T. Tao. Global well-posedness and scattering in the energy space for the critical nonlinear Schrodinger equation
in R3. Annals of Math., (167):767{865, 2008.
[7] A. So er J. Ginibre and G. Velo. The global Cauchy problem for the critical nonlinear wave equation. Jour. Func. Anal., (110):96{130, 1992.
[8] M. Struwe J. Shatah. Well Posedness in the energy space for semilinear wave equations with critical growth. Inter. Math. Research Not., (7):303{309, 1994.
[9] L. Kapitanski. Global and unique weak solutions of nonlinear wave equations. Math. Res. Letters, (1):211{223, 1994.
[10] F. Planchon N. Masmoudi. On Uniqueness for the Critical Wave Equation. Commun. Partial Di erential Equations., 31(7):1099{1107, 2006.
[11] K. Nakanishi. Scattering Theory for Nonlinear Klein-Gordon Equation with Sobolev Critical Power. Internat. Math. Res. Not., (1):31{60, 1999.
[12] K. Nakanishi. Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity. Commun. Partial Di erential Equations., (24):185{221, 1999.
[13] H. Pecher. Nonlinear small data scattering for the wave and Klein-Gordon equations. Math. Z., (185):261{270, 1984.
[14] J. Rauch. The u5 Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations in Nonlinear partial di erential equations and their applications.
[15] I.E. Segal. The global Cauchy problem for a relativistic scalar eld with power interactions. Bull. Soc. Math. France, (91):129{135, 1963.
[16] H.W. Shih. Spacetime bounds for the energy-critical nonlinear wave equation in high spatial dimensions. Ready to submit.
[17] C. D. Sogge. Lectures on Non-linear Wave Equations. Monographs in analysis. International Press, 2008.
[18] M. Struwe. Globally regular solutions to the u5 Klein-Gordon equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (15):495{513, 1988.
[19] T. Tao. Global well-posedness and scattering for the higher-dimensional energycritical nonlinear Schrodinger equation for radial data. New York J. Math., (11):57{80, 2005.
[20] T. Tao. Nonlinear dispersive equations: local and global analysis. Number 106. American Mathematical Soc., 2006.
[21] T. Tao. Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions. Dynamics of PDE, (2):93{110, 2006.