Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity

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Abstract

We study a blow-up curve for the one dimensional wave equation ∂2tu - ∂2xu = |∂tu|p with p > 1. The purpose of this paper is to show that the blow-up curve is a C1 curve if the initial values are large and smooth enough. To prove the result, we convert the equation into a first order system, and then apply a modification of the method of Caffarelli and Friedman [2]. Moreover, we present some numerical investigations of the blow-up curves. From the numerical results, we were able to confirm that the blow-up curves are smooth if the initial values are large and smooth enough. Moreover, we can predict that the blow-up curves have singular points if the initial values are not large enough even they are smooth enough.

Original languageEnglish
Pages (from-to)373-408
Number of pages36
JournalAdvances in Differential Equations
Volume23
Issue number5-6
Publication statusPublished - 2018 May 1
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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