ETD

• critical nonlinear Schrödinger equation
• delta coupling
• finite-time blow-up
• minimal mass
• pseudo-conformal profile
• quantum graphs
• star graphs

We construct finite-time blow-up solutions for the focusing $L^2$-critical quintic nonlinear Schr\"odinger equation on the two-star graph endowed with an arbitrary $\delta$ coupling at the vertex. For every prescribed interior point $a>0$ on one edge, the solutions have exactly the Euclidean minimal mass $M_Q=\frac12\norm[L^2(\R)]{Q}^2$, blow up only at $a$ on the chosen edge, and vanish on the second edge in $H^1(\R_+)$ as the blow-up time is approached. The singular core is a cut-off pseudo-conformal copy of the one-dimensional ground state and is placed away from the vertex; the exponentially small cut-off error is corrected by a terminal fixed-point argument in weighted energy spaces. A second, independent part of the proof shows persistence of the Hamiltonian domain: the constructed solution belongs to $C([0,T),D(H_\alpha))\cap C^1([0,T),L^2(\mathcal G_2))$. This is obtained from strong approximants and time-difference quotient estimates, and ensures that the $\delta$ vertex condition is satisfied by the exact blow-up solution for every pre-blow-up time.