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On the Global Solvability for Semilinear Wave Equations with Smooth Time Dependent Propagation Speeds

  • Fumihiko HirosawaEmail author
  • Takuhiro Inooka
  • Trieu Duong Pham
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 44)

Abstract

In this paper we consider the global existence of a solution with small data to the Cauchy problem for the semilinear wave equations with time dependent coefficient:
$$ u_{tt} - a(t)^2 \varDelta u=u_t^2-a(t)^2| \nabla u|^2. $$
We are particularly interested in the effects of the smoothness to the coefficients in the sense of C m and the Gevrey classes, and the main theorems are natural generalization of the previous results for less smoothness of coefficients.

Keywords

Semi-linear wave equation Cauchy problem Variable coefficient Gevrey class 

Mathematics Subject Classification

35L70 35B40 

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Fumihiko Hirosawa
    • 1
    Email author
  • Takuhiro Inooka
    • 2
  • Trieu Duong Pham
    • 3
  1. 1.Department of Mathematical SciencesYamaguchi UniversityYamaguchiJapan
  2. 2.Graduate School of Science and EngineeringYamaguchi UniversityYamaguchiJapan
  3. 3.Hanoi National University of EducationHanoiVietnam

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