Well-posedness and ill-posedness of the Cauchy problem for the generalized Thirring model. (English) Zbl 1363.35301

Huh, Hyungjin; Machihara, Shuji; Okamoto, Mamoru

Differ. Integral Equ. 29, No. 5-6, 401-420 (2016).

Adapted from the authors’ abstract: We consider the Cauchy problem for the generalized Thirring model in one spatial dimension, that is, \[ (\partial_t \pm \partial_x)U_\pm = i|U_\pm|^k|U_\mp|^{m-k}U_\pm, \]
\[ U_\pm(0, x) = u_\pm(x) \in H^s(\mathbb R). \] Here, \(U_\pm:\mathbb R^{1 + 1} \to \mathbb C\) are unknown functions, \(u_\pm\) are given functions, \(m \in \mathbb N\), and \(k = 0,\dots ,m.\) This model was introduced by H. Huh in the paper [“Global strong solutions to some nonlinear Dirac equations in super-critical space”, Abstr. Appl. Anal. 2013, Article ID 602753, 8 p. (2013)]. Several results concerning well-posedness and ill-posedness are obtained. Since the nonlinearity is not smooth if \(k\) or \(m\) is odd, an upper bound of \(s\) for the problem to be well-posed appears. We prove that the upper bound is essential. More precisely, we show that the problem is ill-posed in \(H^s(\mathbb R)\) for sufficiently large \(s\). This is a novel feature of this paper.

Reviewer: Anthony D. Osborne (Keele)

MSC:

35Q41	Time-dependent Schrödinger equations and Dirac equations
35A01	Existence problems for PDEs: global existence, local existence, non-existence

Keywords:

well-posedness; ill-posedness; Cauchy problem; generalised Thirring model; Sobolev space