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

Abstract

We consider the inhomogeneous nonlinear Schrödinger (INLS) equation in ${\mathbb{R}}^N$ $$i{\partial }_{t}u+\Delta u+|x{|}^{-b}|u{|}^{2\sigma }u=0,$$ where $N\ge 3$, $0<b<\min \left\{\frac{N}{2},2\right\}$ and $\frac{2-b}{N}<\sigma <\frac{2-b}{N-2}$. The scaling invariant Sobolev space is ${\dot{H}}^{s_c}$ with $s_c=\frac{N}{2}-\frac{2-b}{2\sigma }$. The restriction on $\sigma$ implies $0<s_c<1$ and the equation is called intercritical (i.e. mass-supercritical and energy-subcritical). Let $u_0\in {\dot{H}}^{s_c}\cap {\dot{H}}^1$ be a radial initial data and $u(t)$ the corresponding solution to the INLS equation. We first show that if $E[u_0]\le 0$, 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 $T^{\ast }>0$, then ${\mathrm{limsup}}_{t\to T^{\ast }}\Vert u(t){\Vert }_{{\dot{H}}^{s_c}}=+\infty$. Moreover, under an additional assumption and recalling that ${\dot{H}}^{s_c}\subset L^{{\sigma }_c}$ with ${\sigma }_c=\frac{2N\sigma }{2-b}$, we can in fact deduce, for some $\gamma =\gamma (N,\sigma ,b)>0$, the following lower bound for the blow-up rate $$c\|u(t)\|_{\dot H^{s_c}}\geq \|u(t)\|_{L^{σ_c}}\geq |\log (T-t)|^γ,\,\,\,\mbox{ as }\,\,\,t\rightarrow T^{\ast}.$$ The proof is based on the ideas introduced for the $L^2$ super critical nonlinear Schrödinger equation in the work of Merle and Raphaël [13] and here we extend their results to the INLS setting.

References in corpus (3)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• On well-posedness and concentration of blow-up solutions for the intercritical inhomogeneous NLS equation
• The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

Cited by in corpus (2)

• Blow-up of non-radial solutions for the $L^2$ critical inhomogeneous NLS equation
• Interpolation scattering for wave equations with singular potentials and singular data