Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness

Hyungjin Huh

doi:10.3934/eect.2018003

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2018, Volume 7, Issue 1: 53-60. Doi: 10.3934/eect.2018003

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Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness

Hyungjin Huh

Department of Mathematics, Chung-Ang University, Seoul, 156-756, Korea

Received: July 31, 2017

Revised: October 31, 2017

Published: February 2018

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Abstract

Self-similar solutions to nonlinear Dirac systems (1) and (2) are constructed. As an application, we obtain nonuniqueness of strong solution in super-critical space $C([0, T]; H^{s}(\Bbb{R}))$ $(s<0)$ to the system (1) which is $L^2(\Bbb{R})$ scaling critical equations. Therefore the well-posedness theory breaks down in Sobolev spaces of negative order.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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References

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