TY - JOUR

T1 - Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity

AU - Sasaki, Takiko

N1 - Funding Information:
Acknowledgement. The author is grateful to anonymous referees for their valuable comments. I also thank professor Masahito Ohta and professor Norikazu Saito for their valuable suggestions. This work is supported by JST, CREST, and JSPS KAKENHI Grant Number 15H03635, 15K13454.

PY - 2018/5/1

Y1 - 2018/5/1

N2 - 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.

AB - 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.

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M3 - Article

AN - SCOPUS:85040838190

SN - 1079-9389

VL - 23

SP - 373

EP - 408

JO - Advances in Differential Equations

JF - Advances in Differential Equations

IS - 5-6

ER -