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

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 < b < min {N/2, 2} and 2-b/N < sigma < 2-b/N-2. The scaling invariant Sobolev space is (H) over dot(sc) with s(c) = N/2 - 2-b/2 sigma. The restriction on sigma implies 0 < s(c) < 1 and the equation is called intercritical (i.e. mass-supercritical and energy-subcritical). Let u(0) is an element of (H) over dot(sc) boolean AND (H) over dot(1) be a radial initial data and u(t) the corresponding solution to the INLS equation. We first show that if E[u(0)] <= 0, then the maximal time of existence of the solution u(t) is finite. Also, for all radially symmetric solution of the INLS equation with finite maximal time of existence T* > 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.