The Blow-Up Problem for a Class of Quasilinear Schrödinger Equations

We consider the blow-up results of the solution in W2, 2(RN) for the following quasilinear Schrödinger equation $ {align*} {cases} { {array}{ll} iu_t+Δ u+2uh'(|u|^2)Δ h(|u|^2)+uf(|u|^2)=0, x∈ R^N, \\ u(x, 0)=u_0(x), x∈ R^N, {array} } {cases} {align*} $ where $ h $ and $ f $ are real functions which related to various physical models. We prove that the $ W2, 2(R^N) $ solutions must blow up if $ |x|u_0∈ L^2(R^N) $(finite variance), and we give the upper bound of the blow-up time. We also show that without the finite variance assumption, the radial symmetric solutions in $ W2, 2(R^N) $ must blow up in finite time for the whole class of initial data with strictly negative energy.