On Well-Posedness and Concentration of Blow-Up Solutions for the Intercritical Inhomogeneous NLS Equation

We consider the focusing inhomogeneous nonlinear Schrodinger (INLS) equation in R-N i partial derivative(t)u + Delta u + vertical bar x vertical bar(-b)vertical bar u vertical bar(2 sigma) u = 0, where N >= 2 and sigma, b > 0. We first obtain a small data global result in H-1, which, in the two spatial dimensional case, improves the third author result in [22] on the range of b. For N >= 3 and 2-b/N < sigma < 2-b/N-2, we also study the local well posedness in (H)over dot(sc) boolean AND (H)over dot(1), where s(c) = N/2 - 2-b/2 sigma. Sufficient conditions for global existence of solutions in (H)over dot(sc) boolean AND (H)over dot(1) are also established, using a Gagliardo-Nirenberg type estimate. Finally, we study the L-sigma c-norm concentration phenomenon, where sigma(c) = 2N sigma/2-b, for finite time blow-up solutions in (H)over dot(sc) boolean AND (H)over dot(1) with bounded (H)over dot(sc)-norm. Our approach is based on the compact embedding of (H)over dot(sc) boolean AND (H)over dot(1) into a weighted L2 sigma+2 space.