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

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]**Rentaro Agemi and Hiroyuki Takamura,*The lifespan of classical solutions to nonlinear wave equations in two space dimensions*, Hokkaido Math. J.**21**(1992), no. 3, 517–542. MR**1191034**, 10.14492/hokmj/1381413726**[2]**Fumioki 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), no. 13, 1459–1487. MR**862696**, 10.1080/03605308608820470**[3]**J. Ginibre and G. Velo,*Conformal invariance and time decay for nonlinear wave equations. I, II*, Ann. Inst. H. Poincaré Phys. Théor.**47**(1987), no. 3, 221–261, 263–276. MR**921307****[4]**-,*Conformal invariance and time decay for non linear wave equations*. II, Ann. Inst. H. Poincaré Phys. Théor.**47**(1987), 263-276.**[5]**J. Ginibre and G. Velo,*Scattering theory in the energy space for a class of nonlinear wave equations*, Comm. Math. Phys.**123**(1989), no. 4, 535–573. MR**1006294****[6]**Robert T. Glassey,*Finite-time blow-up for solutions of nonlinear wave equations*, Math. Z.**177**(1981), no. 3, 323–340. MR**618199**, 10.1007/BF01162066**[7]**Robert T. Glassey,*Existence in the large for 𝑐𝑚𝑢=𝐹(𝑢) in two space dimensions*, Math. Z.**178**(1981), no. 2, 233–261. MR**631631**, 10.1007/BF01262042**[8]**Fritz John,*Blow-up of solutions of nonlinear wave equations in three space dimensions*, Manuscripta Math.**28**(1979), no. 1-3, 235–268. MR**535704**, 10.1007/BF01647974**[9]**Fritz John,*Nonlinear wave equations, formation of singularities*, University Lecture Series, vol. 2, American Mathematical Society, Providence, RI, 1990. Seventh Annual Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989. MR**1066694****[10]**Sergiu Klainerman,*Long-time behavior of solutions to nonlinear evolution equations*, Arch. Rational Mech. Anal.**78**(1982), no. 1, 73–98. MR**654553**, 10.1007/BF00253225**[11]**Mikhail Kovalyov,*Long-time behaviour of solutions of a system of nonlinear wave equations*, Comm. Partial Differential Equations**12**(1987), no. 5, 471–501. MR**883321**, 10.1080/03605308708820501**[12]**-,*Long-time existence of solutions of nonlinear wave equations*, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, 1986.**[13]**Kôji Kubota,*Existence of a global solution to a semi-linear wave equation with initial data of noncompact support in low space dimensions*, Hokkaido Math. J.**22**(1993), no. 2, 123–180. MR**1226588**, 10.14492/hokmj/1381413170**[14]**Kiyoshi Mochizuki and Takahiro Motai,*The scattering theory for the nonlinear wave equation with small data*, J. Math. Kyoto Univ.**25**(1985), no. 4, 703–715. MR**810974****[15]**Kiyoshi Mochizuki and Takahiro Motai,*The scattering theory for the nonlinear wave equation with small data. II*, Publ. Res. Inst. Math. Sci.**23**(1987), no. 5, 771–790. MR**934671**, 10.2977/prims/1195176032**[16]**Hartmut Pecher,*Scattering for semilinear wave equations with small data in three space dimensions*, Math. Z.**198**(1988), no. 2, 277–289. MR**939541**, 10.1007/BF01163296**[17]**Hartmut Pecher,*Global smooth solutions to a class of semilinear wave equations with strong nonlinearities*, Manuscripta Math.**69**(1990), no. 1, 71–92. MR**1070296**, 10.1007/BF02567913**[18]**Jack Schaeffer,*The equation 𝑢_{𝑡𝑡}-Δ𝑢=\vert𝑢\vert^{𝑝} for the critical value of 𝑝*, Proc. Roy. Soc. Edinburgh Sect. A**101**(1985), no. 1-2, 31–44. MR**824205**, 10.1017/S0308210500026135**[19]**Walter A. Strauss,*Nonlinear scattering theory at low energy*, J. Funct. Anal.**41**(1981), no. 1, 110–133. MR**614228**, 10.1016/0022-1236(81)90063-X**[20]**Kimitoshi Tsutaya,*Global existence theorem for semilinear wave equations with noncompact data in two space dimensions*, J. Differential Equations**104**(1993), no. 2, 332–360. MR**1231473**, 10.1006/jdeq.1993.1076**[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]**Kimitoshi Tsutaya,*A global existence theorem for semilinear wave equations with data of noncompact support in two space dimensions*, Comm. Partial Differential Equations**17**(1992), no. 11-12, 1925–1954. MR**1194745**, 10.1080/03605309208820909

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