Local well-posedness for nonlinear Dirac equation on $N$-star metric graphs

We consider the Cauchy problem for the nonlinear Dirac equation on a noncompact $N$-star metric graph $G$, $$\mathrm{i}\partial_t \psi = D\psi - |\psi|^{p-2}\psi, \qquad \psi(0)=\psi_0,$$ where $p\ge3$, $\psi:\mathbb{R}\times G\to\mathbb{C}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. Using Bourgain-type spaces defined through the spectral resolution of $D$, together with elementary $L^\infty$ bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well-posedness for initial data $$\psi_0\in H_D^s(G)\cap L^\infty(G;\mathbb C^2), \qquad 0\le s<\frac12 .$$ The corresponding solution belongs to $$C([0,T];H_D^s(G))\cap X_T^{s,b}\cap L^\infty([0,T]\times G).$$ Moreover, $\|\psi(t)\|_{L^2(G;\mathbb{C}^2)}$ is conserved along the solution on the existence interval. We also establish a blow-up alternative in the combined $H_D^s$ and space-time $L^\infty$ control norm.