On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities

By Binhua Feng. 2018

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This paper is devoted to the analysis of blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities $$\begin{aligned} iu_{t}+\Delta u=\lambda _1|u|^{p_1}u+\lambda _2|u|^{p_2}u. \end{aligned}$$iut+Δu=λ1|u|p1u+λ2|u|p2u.When $$p_1=\frac{4}{N}$$p1=4N and $$0<p_2<\frac{4}{N}$$0<p2<4N, we prove the existence of blow-up solutions and find the sharp threshold mass of blow-up and global existence for this equation. This is a complement to the result of Tao et al. (Commun Partial Differ Equ 32:1281–1343, 2007). Moreover, we investigate the dynamical properties of blow-up solutions, including $$L^2$$L2-concentration, blow-up rates and limiting profile. When $$\frac{4}{N}<p_1<\frac{4}{N-2}$$4N<p1<4N-2($$4<p_1<\infty $$4<p1<∞ if $$N=1$$N=1, $$2<p_1<\infty $$2<p1<∞ if $$N=2$$N=2), we prove that the blow-up solution with bounded $$\dot{H}^{s_c}$$H˙sc-norm must concentrate at least a fixed amount of the $$\dot{H}^{s_c}$$H˙sc-norm and, also, its $$L^{p_c}$$Lpc-norm must concentrate at least a fixed $$L^{p_c}$$Lpc-norm.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/954c10b4c58ffb9b897142b4e9d6f98fe5a0e369

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