Blow-up of radial solutions for the intercritical inhomogeneous NLS equation

摘要

We consider the inhomogeneous nonlinear Schrodinger (INLS) equation in R-N i partial derivative tu + Delta u + vertical bar x vertical bar(-b)vertical bar u vertical bar(2 sigma)u = 0, where N >= 3, 0 0, then lim sup(t -> T*) vertical bar vertical bar u(t)vertical bar vertical bar((H) over dotsc) = +infinity. Moreover, under an additional assumption and recalling that (H) over dot(sc) subset of L-sigma c with sigma(c) = 2N sigma/2-b, we can in fact deduce, for some gamma = gamma(N, sigma, b) > 0, the following lower bound for the blow-up rate c parallel to u(t)parallel to((H) over dotsc) >= parallel to u(t)parallel to(L sigma c) >= vertical bar log(T - t)vertical bar gamma, as t -> T*. The proof is based on the ideas introduced for the L-2 super critical nonlinear Schrodinger equation in the work of Merle and Raphael [14] and here we extend their results to the INLS setting. (C) 2021 Elsevier Inc. All rights reserved.