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The blow-up curve of solutions to one dimensional nonlinear wave equations with the Dirichlet boundary conditions

  • Tetsuya Ishiwata
  • Takiko SasakiEmail author
Original Paper
  • 16 Downloads

Abstract

In this paper, we consider the blow-up curve of semilinear wave equations. Merle and Zaag (Am J Math 134:581–648, 2012) considered the blow-up curve for \(\partial _t^2 u- \partial _x^2 u = |u|^{p-1}u\) and showed that there is the case that the blow-up curve is not differentiable at some points when the initial value changes its sign. Their analysis depends on the variational structure of the problem. In this paper, we consider the blow-up curve for \(\partial _t^2 u- \partial _x^2 u = |\partial _t u|^{p-1}\partial _t u\) which does not have the variational structure. Nevertheless, we prove that the blow-up curve is not differentiable if the initial data changes its sign and satisfies some conditions.

Keywords

Blow-up Wave equation Positive solutions 

Mathematics Subject Classification

35B44 35L05 35B09 

Notes

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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesShibaura Institute of TechnologySaitamaJapan
  2. 2.National Institute of Technology, Ibaraki CollegeHitachinakaJapan

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