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Decay estimates for the critical semilinear wave equation. (English) Zbl 0924.35084
Authors’ abstract: In this paper we prove that finite energy solutions (with added regularity) to the critical wave equation \(u_{tt}-\Delta u+u^5=0\) on \(\mathbb{R}^3\) decay to zero in time. The proof is based on a global space-time estimate and dilation identity.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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References:
[1] {\scH. Bahouri, P. Gerard}, Private communications.
[2] Ginibre, J.; Soffer, A.; Velo, G., The global Cauchy problem for the critical nonlinear wave equation, J. func. analysis, Vol. 110, 96-130, (1992) · Zbl 0813.35054
[3] Ginibre, J.; Velo, G., The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. zeit., Vol. 189, 487-505, (1985) · Zbl 0549.35108
[4] Grillakis, M., Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of math., Vol. 132, 4509-4875, (1990)
[5] Jörgen, K., Das anfangswertproblem in grossen für eine klasse nichtlinearer wellengleichunger, Math. zeit., Vol. 77, 295-308, (1961)
[6] {\scJ. Rauch}, The u5 Klein-Gordon equation, Nonlinear PDE’s and Applications, Pitman Research Notes in Math., Vol. {\bf53}, pp. 335-364.
[7] Segal, I.E., The global Cauchy problem for a relativistic scalar field with power interaction, Bull. soc. math. France, Vol. 91, 129-135, (1963) · Zbl 0178.45403
[8] Shatah, J.; Struwe, M., Regularity results for nonlinear wave equation, Ann. of math., Vol. 138, 503-518, (1993) · Zbl 0836.35096
[9] Shatah, J.; Struwe, M., Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. math. res. notices, Vol. 7, 303-309, (1994) · Zbl 0830.35086
[10] Strauss, W., Decay and asymptotics for □u = F(u), J. funct. anal., Vol. 2, 405-457, (1968) · Zbl 0182.13602
[11] Strauss, W., Nonlinear invariant wave equations, (), 197-249
[12] Strauss, W.A., Nonlinear scattering theory of low energy, J. funct. anal., Vol. 41, 110-133, (1981) · Zbl 0466.47006
[13] Strichartz, R.S., Convolutions with kernels having singularities on a sphere, Trans. A.M.S., Vol. 148, 461-471, (1970) · Zbl 0199.17502
[14] Strichartz, R.S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke math. J., Vol. 44, 705-714, (1977) · Zbl 0372.35001
[15] Struwe, M., Globally regular solutions to the u5 Klein-Gordon equation, Ann. scuola norm. sup. Pisa cl. sci., Vol. 15, 4, 495-513, (1988) · Zbl 0728.35072
[16] Zuily, C., Solutions en grand temps d’ equations d’ ondes non linéares Séminaire boubaki, (), n^{o} 779
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