Minimal Mass Blow-Up Solutions for the \ (L^2\) -Critical NLS with the Delta Potential for Even Data in One Dimension

. We consider the \ (L²\) -critical nonlinear Schrödinger equation (NLS) with the delta potential \ (iₜu +²ₓ u + u +|u|^4u=0, \, \, t R, \, x R, \) where \ (R\) and \ (\) is the Dirac delta distribution at \ (x=0\). Local well-posedness theory, together with the 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. , \ (=0\) ). We are interested in the dynamics of the solution with threshold mass \ (\|u₀\|₂=\|Q\|₂\) in \ (H¹ (R) \). First, for the case \ (=0\), such a blow-up solution exists due to the pseudoconformal symmetry of the equation and is unique up to the symmetries of the equation in \ (H¹ (R) \) from Merle Duke Math. J. , 69 (1993), pp. 427–454 and recently in \ (L² (R) \) from Dodson arXiv: 2104. 11690, 2021. Second, for the case \ (0\), a simple variational argument with the conservation laws of mass and energy implies that even solutions with threshold mass exist globally in \ (H¹ (R) \). Finally, for the case \ (0\), we show the existence of even threshold solutions with blow-up speed determined by the sign (i. e. , \ (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 the compactness method as well as modulation analysis, are close to the original one in Raphaël and Szeftel J. Amer. Math. Soc. , 24 (2011), pp. 471–546. Keywordsblow-upconcentration-compactness argumentcompactness methodDirac delta potentialEnergy–Morawetz estimatemodulation analysisnonlinear Schrödinger equationMSC codes35Q5535B44