Low-regularity well-posedness for a mixed-sign quadratic Dirac equation on N-star metric graphs

Abstract

We study the Cauchy problem for a mixed-sign quadratic Dirac equation on a noncompact N–star metric graph G, (Formula presented.) where ψ=(ψ1,ψ2)T:R×G→C2 and D denotes the self-adjoint Dirac–Kirchhoff operator on G. The nonlinearity acts edgewise and is given by a bilinear interaction between the positive and negative spectral parts, (Formula presented.) where Π± are the spectral projections of D and B is a fixed bilinear map on C2 applied componentwise on each edge. This is a model quadratic interaction tailored to the mixed-sign Bourgain-space mechanism, rather than a general nonlinear Dirac equation on graphs. Using Bourgain-type spaces associated with the spectral resolution of D and a mixed-sign bilinear estimate on N–star graphs, we prove local well-posedness in the operator Sobolev space HDs(G) for s>-18. We also establish a blow-up alternative in HDs(G) for the maximal forward lifespan.