Advances in Differential Equations

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

Takiko Sasaki

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Abstract

We study a blow-up curve for the one dimensional wave equation $\partial_t^2 u- \partial_x^2 u = |\partial_t u|^p$ with $p>1$. The purpose of this paper is to show that the blow-up curve is a $C^1$ 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.

Article information

Source
Adv. Differential Equations Volume 23, Number 5/6 (2018), 373-408.

Dates
First available in Project Euclid: 23 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.ade/1516676482

Subjects
Primary: 35L15: Initial value problems for second-order hyperbolic equations 35L67: Shocks and singularities [See also 58Kxx, 76L05]

Citation

Sasaki, Takiko. Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity. Adv. Differential Equations 23 (2018), no. 5/6, 373--408. https://projecteuclid.org/euclid.ade/1516676482


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