Scattering theory for semilinear wave equations with small data in two space dimensions
Author:
Kimitoshi Tsutaya
Journal:
Trans. Amer. Math. Soc. 342 (1994), 595618
MSC:
Primary 35P25; Secondary 35L70, 35P30, 47F05, 47N20
MathSciNet review:
1214786
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Abstract: We study scattering theory for the semilinear wave equation in two space dimensions. We show that if , the scattering operator exists for smooth and small data. The lower bound of p is considered to be optimal (see Glassey [6, 7], Schaeffer [18]). Our result is an extension of the results by Strauss [19], Klainerman [10], and Mochizuki and Motai [14, 15]. The construction of the scattering operator for small data does not follow directly from the proofs in [7, 13, 20 and 22] concerning the global existence of solutions for the Cauchy problem of the above equation with small initial data given at in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation: for , where is a solution of which approaches asymptotically as . The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of in two space dimensions.
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, Global existence and the life span of solutions of semilinear wave equations with data of non compact support in three space dimensions, (to appear).
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 [1]
 R. Agemi and H. Takamura, The lifespan of classical solutions to nonlinear wave equations in two space dimensions, Hokkaido Math. J. 21 (1992), 517542. MR 1191034 (93m:35126)
 [2]
 F. Asakura, Existence of a global solution to a semilinear wave equation with slowly decreasing initial data in three space dimensions, Comm. Partial Differential Equations 11 (1986), 14591487. MR 862696 (87k:35165)
 [3]
 J. Ginibre and G. Velo, Conformal invariance and time decay for non linear wave equations. I, Ann. Inst. H. Poincaré Phys. Théor. 47 (1987), 221261. MR 921307 (89e:35021)
 [4]
 , Conformal invariance and time decay for non linear wave equations. II, Ann. Inst. H. Poincaré Phys. Théor. 47 (1987), 263276.
 [5]
 , Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys. 123 (1989), 535573. MR 1006294 (90i:35172)
 [6]
 R. T. Glassey, Finitetime blowup for solutions of nonlinear wave equations, Math. Z. 177 (1981), 323340. MR 618199 (82i:35120)
 [7]
 , Existence in the large for in two space dimensions, Math. Z. 178 (1981), 233261. MR 631631 (84h:35106)
 [8]
 F. John, Blowup of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235268. MR 535704 (80i:35114)
 [9]
 , Nonlinear wave equations, formation of singularities, Univ. Lecture Ser., Amer. Math. Soc., Providence, RI, 1990. MR 1066694 (91g:35001)
 [10]
 S. Klainerman, Longtime behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal. 78 (1982), 7398. MR 654553 (84b:35015)
 [11]
 M. Kovalyov, Longtime behaviour of solutions of a system of nonlinear wave equations, Comm. Partial Differential Equations 12 (1987), 471501. MR 883321 (88c:35100)
 [12]
 , Longtime existence of solutions of nonlinear wave equations, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, 1986.
 [13]
 K. Kubota, Existence of a global solution to a semilinear wave equation with initial data of noncompact support in low space dimensions, Hokkaido Math. J. 22 (1993), 123180. MR 1226588 (94f:35088)
 [14]
 K. Mochizuki and T. Motai, The scattering theory for the nonlinear wave equation with small data, J. Math. Kyoto Univ. 25 (1985), 703715. MR 810974 (87i:35121)
 [15]
 , The scattering theory for the nonlinear wave equation with small data. II, Publ. Res. Inst. Math. Sci. 23 (1987), 771790. MR 934671 (89f:35138)
 [16]
 H. Pecher, Scattering for semilinear wave equations with small data in three space dimensions, Math. Z. 198 (1988), 277289. MR 939541 (89e:35123)
 [17]
 , Global smooth solutions to a class of semilinear wave equations with strong nonlinearities, Manuscripta Math. 69 (1990), 7192. MR 1070296 (91f:35174)
 [18]
 J. Schaeffer, The equation for the critical value of p, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 3144. MR 824205 (87g:35159)
 [19]
 W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1981), 110133. MR 614228 (83b:47074a)
 [20]
 K. Tsutaya, Global existence theorem for semilinear wave equations with noncompact data in two space dimensions, J. Differential Equations 104 (1993), 332360. MR 1231473 (94g:35152)
 [21]
 , Global existence and the life span of solutions of semilinear wave equations with data of non compact support in three space dimensions, (to appear).
 [22]
 , A global existence theorem for semilinear wave equations with data of non compact support in two space dimensions, Comm. Partial Differential Equations 17 (1992), 19251954. MR 1194745 (93m:35120)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412147861
PII:
S 00029947(1994)12147861
Keywords:
Scattering theory,
semilinear wave equations,
two space dimensions
Article copyright:
© Copyright 1994
American Mathematical Society
