2021/10/15 by Tang, Xingdong, Xu, Guixiang
#35B44 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2110.07834
We consider the L2-critical nonlinear Schrödinger equation (NLS) with the delta potential i∂tu +∂2x u + μδu +|u|4u=0, t∈ \R, x∈ \R , where μ∈ \R, and δ is the Dirac delta distribution at x=0. Local well-posedness theory together with sharp Gagliardo-Nirenberg inequality and the conservation laws of mass and energy implies that the solution with mass less than ‖Q‖2 is global existence in H1(\R), where Q is the ground state of the L2-critical NLS without the delta potential (i.e. μ=0). We are interested in the dynamics of the solution with threshold mass ‖u0‖2=‖Q‖2 in H1(\R). First, for the case μ=0, such blow-up solution exists due to the pseudo-conformal symmetry of the equation, and is unique up to the symmetries of the equation in H1(\R) from \citeMe93:NLS:mini sol (see also \citeHmKe05:NLS:mini blp), and recently in L2(\R) from \citeDod:NLS:L2thrh1. Second, for the case μ<0, simple variational argument with the conservation laws of mass and energy implies that radial solutions with threshold mass exist globally in H1(\R). Last, for the case μ>0, we show the existence of radial threshold solutions with blow-up speed determined by the sign (i.e. μ>0) of the delta potential perturbation since the refined blow-up profile to the rescaled equation is stable in a precise sense. The key ingredients here including the Energy-Morawetz argument and compactness method as well as the modulation analysis are close to the original one in \citeRaS11:NLS:mini sol (see also \citeKrLR13:HalfW:nondis, LeMR:CNLS:blp, Mart05:Kdv:N sol, MaP17:BO:mini sol, MeRS14:NLS:blp).