Limit of the blow-up solution for the inhomogeneous nonlinear Schr?dinger equation?

We study the H1 blow-up profile for the inhomogeneous nonlinear Schrodinger equation i partial differential tu = - increment u - |x|k|u|2 sigma u, (t, x) is an element of R x RN, where k is an element of (-1, 2N - 2) and N >= 3. We develop a new version of Gagliardo-Nirenberg inequality for sigma is an element of [2+k N , 2+k N -2 ] and k is an element of (-1, 2N - 2), and show that for the L2-critical exponent sigma = 2+k N , u(t) has no L2-limit as t -> T* when parallel to u(t)parallel to H1 blows up at T*. Moreover, we investigate L2 concentration at the origin in the radial case. Additionally, if 2+k N < sigma < min{2, 2+k N-2, 2(1+k)+N 2N }, we show that there exists a unique u* is an element of L2(RN) such that Gamma(-t)u(t) -> Gamma(-T*)u* in Lr(RN) (r is an element of [2, 2*)) as t -> T*. Our results extend the work for k = 0 by F. Merle in earlier time. (c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.