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We consider the nonlinear Schr\"odinger equations on the star graph with theKirchhoff boundary and the repulsive Dirac delta boundary at the origin. In thepresent paper, we show the scattering-blowup dichotomy result below themass-energy of the ground state on the real line. The proof of the scatteringpart is based on a concentration compactness and rigidity argument. Our maincontribution is to give a linear profile decomposition on the star graph byusing a symmetrical decomposition.

Global dynamics below a threshold for the nonlinear Schr\"odinger  equations with the Kirchhoff boundary and the repulsive Dirac delta boundary  on a star graph