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 $G$, $$\mathrm{i}{\mathrm{\partial }}_t\psi =D\psi -\mathrm{\mid }\psi {\mathrm{\mid }}^{p-2}\psi ,\psi (0)={\psi }_0,$$ where $p\ge 3$, $\psi :\mathbb{R}\times G\to {\mathbb{C}}^2$ 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 $L^{\mathrm{\infty }}$ bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well-posedness for initial data $${\psi }_0\in H_D^s(G)\cap L^{\mathrm{\infty }}(G;{\mathbb{C}}^2),0\le s<\frac{1}{2}\mathrm{.}$$ The corresponding solution belongs to $$C([0,T];H_D^s(G))\cap X_T^{s,b}\cap L^{\mathrm{\infty }}([0,T]\times G)\mathrm{.}$$ Moreover, $\mathrm{\parallel }\psi (t){\mathrm{\parallel }}_{L^2(G;{\mathbb{C}}^2)}$ 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 $L^{\mathrm{\infty }}$ control norm.