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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Scattering theory for semilinear wave equations with small data in two space dimensions
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by Kimitoshi Tsutaya PDF
Trans. Amer. Math. Soc. 342 (1994), 595-618 Request permission

Abstract:

We study scattering theory for the semilinear wave equation ${u_{tt}} - \Delta u = |u{|^{p - 1}}u$ 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: \[ u(x,t) = u_0^ - (x,t) + \frac {1}{{2\pi }}\int _{ - \infty }^t {\int _{|x - y| \leq t - s} {\frac {{(|u{|^{p - 1}}u)(y,s)}}{{\sqrt {{{(t - s)}^2} - |x - y{|^2}} }}dy\;ds,} } \] 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 ${u_{tt}} - \Delta u = |u{|^{p - 1}}u$ in two space dimensions.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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