Dynamics of Partial Differential Equations

Volume 13 (2016)

Number 3

Remarks on ill-posedness for the Dirac–Klein–Gordon system

Pages: 179 – 190

DOI: https://dx.doi.org/10.4310/DPDE.2016.v13.n3.a1

Authors

Machihara Shuji (Department of Mathematics, Faculty of Science, Saitama University, Sakura-ku, Saitama City, Japan)

Okamoto Mamoru (Department of Mathematics, Institute of Engineering, Academic Assembly, Shinshu University, Wakasato, Nagano City, Japan)

Abstract

We show ill-posedness of the Cauchy problem for the Dirac–Klein–Gordon system in one spatial dimension with some indices of the Sobolev spaces which the initial data belong to. By combining with the existing papers, we define the entire range of those indices for well-posedness or ill-posedness with the exception of one point. At this point, it is still unsolved whether well-posedness holds or not with respect to Sobolev spaces. We introduce one solvability for the problem of this point by giving the result of the unique existence of solution in the corresponding Lebesgue spaces.

Keywords

Dirac–Klein–Gordon system, Cauchy problem, well-posedness, ill-posedness

2010 Mathematics Subject Classification

35B30, 35Q41, 35R25

Published 23 June 2016