Scattering theory for semilinear wave equations with small data in two space dimensions

Author:
Kimitoshi Tsutaya

Journal:
Trans. Amer. Math. Soc. **342** (1994), 595-618

MSC:
Primary 35P25; Secondary 35L70, 35P30, 47F05, 47N20

DOI:
https://doi.org/10.1090/S0002-9947-1994-1214786-1

MathSciNet review:
1214786

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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:

**[1]**R. Agemi and H. Takamura,*The lifespan of classical solutions to nonlinear wave equations in two space dimensions*, Hokkaido Math. J.**21**(1992), 517-542. MR**1191034 (93m:35126)****[2]**F. Asakura,*Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions*, Comm. Partial Differential Equations**11**(1986), 1459-1487. 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), 221-261. MR**921307 (89e:35021)****[4]**-,*Conformal invariance and time decay for non linear wave equations*. II, Ann. Inst. H. Poincaré Phys. Théor.**47**(1987), 263-276.**[5]**-,*Scattering theory in the energy space for a class of non-linear wave equations*, Comm. Math. Phys.**123**(1989), 535-573. MR**1006294 (90i:35172)****[6]**R. T. Glassey,*Finite-time blow-up for solutions of nonlinear wave equations*, Math. Z.**177**(1981), 323-340. MR**618199 (82i:35120)****[7]**-,*Existence in the large for**in two space dimensions*, Math. Z.**178**(1981), 233-261. MR**631631 (84h:35106)****[8]**F. John,*Blow-up of solutions of nonlinear wave equations in three space dimensions*, Manuscripta Math.**28**(1979), 235-268. 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,*Long-time behavior of solutions to nonlinear evolution equations*, Arch. Rational Mech. Anal.**78**(1982), 73-98. MR**654553 (84b:35015)****[11]**M. Kovalyov,*Long-time behaviour of solutions of a system of nonlinear wave equations*, Comm. Partial Differential Equations**12**(1987), 471-501. MR**883321 (88c:35100)****[12]**-,*Long-time 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 semi-linear wave equation with initial data of non-compact support in low space dimensions*, Hokkaido Math. J.**22**(1993), 123-180. 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), 703-715. MR**810974 (87i:35121)****[15]**-,*The scattering theory for the nonlinear wave equation with small data*. II, Publ. Res. Inst. Math. Sci.**23**(1987), 771-790. MR**934671 (89f:35138)****[16]**H. Pecher,*Scattering for semilinear wave equations with small data in three space dimensions*, Math. Z.**198**(1988), 277-289. MR**939541 (89e:35123)****[17]**-,*Global smooth solutions to a class of semilinear wave equations with strong nonlinearities*, Manuscripta Math.**69**(1990), 71-92. MR**1070296 (91f:35174)****[18]**J. Schaeffer,*The equation**for the critical value of p*, Proc. Roy. Soc. Edinburgh Sect. A**101**(1985), 31-44. MR**824205 (87g:35159)****[19]**W. A. Strauss,*Nonlinear scattering theory at low energy*, J. Funct. Anal.**41**(1981), 110-133. MR**614228 (83b:47074a)****[20]**K. Tsutaya,*Global existence theorem for semilinear wave equations with non-compact data in two space dimensions*, J. Differential Equations**104**(1993), 332-360. 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), 1925-1954. MR**1194745 (93m:35120)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35P25,
35L70,
35P30,
47F05,
47N20

Retrieve articles in all journals with MSC: 35P25, 35L70, 35P30, 47F05, 47N20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1214786-1

Keywords:
Scattering theory,
semilinear wave equations,
two space dimensions

Article copyright:
© Copyright 1994
American Mathematical Society