Local Well-Posedness for Nonlinear Dirac Equation on N-Star Metric Graphs

Abstract

We consider the Cauchy problem for the nonlinear Dirac equation on a noncompact $N$-star metric graph $G$,

$i{\partial }_t\psi =D\psi -{\left|\psi \right|}^{p-2}\psi ,\psi \left(0\right)={\psi }_0,$

where $p\ge 3$, $\psi :R\times G\to C^2$, 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

The corresponding solution belongs to

$C([0,T];H_D^s\left(G\right))\cap X_T^{s,b}\cap L^{\infty }([0,T]\times G).$

Moreover, ${\parallel \psi \left(t\right)\parallel }_{L^2(G;C^2)}$ 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.