Conformal conservation law, time decay and scattering for nonlinear wave equations

Journal d'Analyse Mathematique - Tập 91 - Trang 269-295 - 2003
Kunio Hidano1
1Department of Mathematics, Tokyo Metropolitan University, Tokyo, Japan

Tóm tắt

We study the implications of the conformal conservation law for the time decay of solutions of nonlinear wave equations and present some improvements over previous work of Ginibre and Velo. We also consider the theory of nonlinear scattering and prove asymptotic completeness in a weighted Sobolev space.

Tài liệu tham khảo

J. C. Baez, I. E. Segal and Z.-F. Zhou,The global Goursat problem and scattering for nonlinear wave equations, J. Funct. Anal.93 (1990), 239–269.

J. Bergh and J. Löfström,Interpolation Spaces, Springer-Verlag, Berlin-Heidelberg, 1976.

J. Ginibre,Scattering theory in the energy space for a class of non-linear wave equations, inSpectral and Scattering Theory and Application, Adv. Stud. Pure Math.23, Kinokuniya, Tokyo, 1994, pp. 83–103.

J. Ginibre and G. Velo,The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z.189 (1985), 487–505.

J. Ginibre and G. Velo,Conformal invariance and time decay for nonlinear wave equations. I, Ann. Inst. Henri Poincaré, Phys. Théor.47 (1987), 221–261.

J. Ginibre and G. Velo,Conformal invariance and time decay for nonlinear wave equations. II, Ann. Inst. Henri Poincaré, Phys. Théor.47 (1987), 263–276.

J. Ginibre and G. Velo,Scattering theory in the energy space for a class of non-linear wave equations, Comm. Math. Phys.123 (1989), 535–573.

J. Ginibre and G. Velo,Generalized Strichartz inequalities for the wave equation, J. Funct. Anal.133 (1995), 50–68.

N. Hayashi and Y. Tsutsumi,Scattering theory for Hatree type equations, Ann. Inst. Henri Poincaré, Phys. Théor.46 (1987), 187–213.

K. Hidano,Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcial. Ekvac.43 (2000), 559–588.

K. Hidano,Scattering problem for the nonlinear wave equation in the finite energy and conformal charge space, J. Funct. Anal.187 (2001), 274–307.

K. Hidano,Scattering and self-similar solutions for the nonlinear wave equation, Differential Integral Equations15 (2002), 405–462.

M. Keel and T. Tao,Endpoint Strichartz estimates, Amer. J. Math.120 (1998), 955–980.

S. Klainerman,Uniform decay estimate and the Lorentz invariance of the classical wave equations, Comm. Pure Appl. Math.38 (1985), 321–332.

S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space ℂn+1, Comm. Pure Appl. Math.40 (1987), 111–117.

T. T. Li and X. Yu,Life-span of classical solutions to fully nonlinear wave equations, Comm. Partial Differential Equations16 (1991), 909–940.

T. T. Li and Y. Zhou,A note on the life-span of classical solutions to nonlinear wave equations in four space dimensions, Indiana Univ. Math. J.44 (1995), 1207–1248.

H. Lindblad and C. D. Sogge,On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal.130 (1995), 357–426.

K. Mochizuki and T. Motai,The scattering theory for the nonlinear wave equation, J. Math. Kyoto Univ.25 (1985), 703–715.

K. Mochizuki and T. Motai,The scattering theory for the nonlinear wave equation II, Publ. Res. Inst. Math. Sci.23 (1987), 771–790.

C. S. Morawetz,The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math.14 (1961), 561–568.

C. S. Morawetz,The limiting amplitude principle, Comm. Pure Appl. Math.15 (1962), 349–361.

K. Nakanishi,Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity, Comm. Partial Differential Equations24 (1999), 185–221.

H. Pecher,Decay of solutions of nonlinear wave equations in three space dimensions, J. Funct. Anal.46 (1982), 221–229.

H. Pecher,Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z.185 (1984), 261–270.

J. Shatah and M. Struwe,Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices (1994), 303–310.

C. D. Sogge,Lectures on Nonlinear Wave Equations, International Press, Cambridge, MA, 1995.

W. A. Strauss,Decay and asymptotics for ▭u=F(u), J. Funct. Anal.2 (1968), 409–457.

W. A. Strauss,Nonlinear scattering theory at low energy, J. Funct. Anal.41 (1981), 110–133.

R. S. Strichartz,Restrictions of Fourier Transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J.44 (1977), 705–714.

Y. Zhou,Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Partial Differential Equations8 (1995), 135–144.

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Journal d'Analyse Mathematique

Tập 91

269-295

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