A blow-up result for the travelling waves of the pseudo-relativistic Hartree equation with small velocity

In this paper, we consider the pseudo-relativistic Hartree equation i partial derivative(t)psi= (root-Delta+m(2)) psi - (1/vertical bar x vertical bar * vertical bar psi vertical bar(2)) psi on R-3 and study travelling solitary waves of the form psi(t, x) = e(it mu)phi(x - vt), where v is an element of R-3 denotes travelling velocity. Frohlich, Jonsson and Lenzmann in [Comm. Math. Phys. 2007, 274:1-30] proved that for vertical bar v vertical bar there exists a critical constant N-c(v), such that the travelling waves exist if and only if 0 < N < N-c(v), where N denotes particle number. In this paper, we consider v=(beta,0,0) with 0 < beta < 1, and let N-c(beta)=N-c(v)vertical bar(v=(beta,0,0)). We find that N-c(beta) is Lipschitz continuity with respect to beta. Based on this fact, we then prove that the boosted ground states phi(beta) with parallel to phi(beta)parallel to(2)(L2) = (1-beta)N-c(beta) satisfy lim(beta -> 0+) parallel to phi(beta) parallel to(H1/2) -> + infinity. The explicit blow-up profile and rate will be computed.