Scattering theory for semilinear wave equations with small data in two space dimensions
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- by Kimitoshi Tsutaya PDF
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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.References
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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