Differential and Integral Equations

Well-posedness and ill-posedness of the Cauchy problem for the generalized Thirring model

Hyungjin Huh, Shuji Machihara, and Mamoru Okamoto

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Abstract

We consider the Cauchy problem for the generalized Thirring model $(\partial _t \pm \partial _x ) U_{\pm} = i |U_{\pm}|^k |U_{\mp}|^{m-k} U_{\pm}$ in one spatial dimension which was introduced in [4]. Several results on well-posedness and ill-posedness have been obtained. Since the nonlinearity is not smooth if $k$ or $m$ is odd, an upper bound of $s$ to be well-posed appears. We prove that the upper bound is essential. More precisely, we show ill-posedness in $H^s(\mathbb{R})$ for sufficiently large $s$ which is a novel feature of this paper.

Article information

Source
Differential Integral Equations Volume 29, Number 5/6 (2016), 401-420.

Dates
First available in Project Euclid: 9 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.die/1457536884

Mathematical Reviews number (MathSciNet)
MR3471966

Zentralblatt MATH identifier
06562182

Subjects
Primary: 35Q41: Time-dependent Schrödinger equations, Dirac equations 35A01: Existence problems: global existence, local existence, non-existence

Citation

Huh, Hyungjin; Machihara, Shuji; Okamoto, Mamoru. Well-posedness and ill-posedness of the Cauchy problem for the generalized Thirring model. Differential Integral Equations 29 (2016), no. 5/6, 401--420. https://projecteuclid.org/euclid.die/1457536884.


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