Convergence of a blow-up curve for a semilinear wave equation

Takiko Sasaki

doi:10.3934/dcdss.2020388

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2021, Volume 14, Issue 3: 1133-1143. Doi: 10.3934/dcdss.2020388

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Convergence of a blow-up curve for a semilinear wave equation

Takiko Sasaki

National Institute of Technology, Ibaraki College, 866 Nakane, Hitachinaka-shi, Ibaraki-ken 312-8508, Japan

Received: December 31, 2018

Revised: January 31, 2020

Early access: June 2020

Published: March 2021

This work was supported by JSPS Grant-in-Aid for Early-Career Scientists, 18K13447

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Abstract

We consider a blow-up phenomenon for $ { \partial_t^2 u_ \varepsilon} $ $ {- \varepsilon^2 \partial_x^2u_ \varepsilon } $ $ { = F(\partial_t u_ \varepsilon)}. $ The derivative of the solution $ \partial_t u_ \varepsilon $ blows-up on a curve $ t = T_ \varepsilon(x) $ if we impose some conditions on the initial values and the nonlinear term $ F $. We call $ T_ \varepsilon $ blow-up curve for $ { \partial_t^2 u_ \varepsilon} $ $ {- \varepsilon^2 \partial_x^2u_ \varepsilon } $ $ { = F(\partial_t u_ \varepsilon)}. $ In the same way, we consider the blow-up curve $ t = \tilde{T}(x) $ for $ {\partial_t^2 u} $ $ = $ $ {F(\partial_t u)}. $ The purpose of this paper is to show that, for each $ x $, $ T_ \varepsilon(x) $ converges to $ \tilde{T}(x) $ as $ \varepsilon\rightarrow 0. $

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Mathematics Subject Classification: Primary: 35B44; Secondary: 35L05, 65M06.

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