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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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
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Abstract: We study scattering theory for the semilinear wave equation $ {u_{tt}} - \Delta u = \vert u{\vert^{p - 1}}u$ in two space dimensions. We show that if $ p > {p_0} = (3 + \sqrt {17} )/2$, the scattering operator exists for smooth and small data. The lower bound $ {p_0}$ 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 $ t = 0$ in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation:

$\displaystyle u(x,t) = u_0^ - (x,t) + \frac{1}{{2\pi }}\int_{ - \infty }^t {\in... ...t^{p - 1}}u)(y,s)}}{{\sqrt {{{(t - s)}^2} - \vert x - y{\vert^2}} }}dy\;ds,} } $

for $ t \in R$, where $ u_0^ - (x,t)$ is a solution of $ {u_{tt}} - \Delta u = 0$ which $ u(x,t)$ approaches asymptotically as $ t \to - \infty $. The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of $ {u_{tt}} - \Delta u = \vert u{\vert^{p - 1}}u$ in two space dimensions.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1214786-1
PII: S 0002-9947(1994)1214786-1
Keywords: Scattering theory, semilinear wave equations, two space dimensions
Article copyright: © Copyright 1994 American Mathematical Society