Finite time blow-up of non-radial solutions for some inhomogeneous Schrödinger equations

2023/06/27 by Bai, Ruobing, Saanouni, Tarek
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.15210

Abstract

This work studies the inhomogeneous Schrödinger equation i∂_t u-K_s,λu +F(x,u)=0 , u(t,x):ℝ×ℝ^N→ℂ. Here, s∈\1,2\, N>2s and λ>-((N-2)²)/(4). The linear Schrödinger operator reads K_s,λ:= (-Δ)^s +(2-s)\fracλ|x|² and the focusing source term is local or non-local F(x,u)∈\|x|^-2τ|u|^2(q-1)u,|x|^-τ|u|^p-2(J_α*|⋅|^-τ|u|^p)u\. The Riesz potential is J_α(x)=C_N,α|x|^-(N-α), for certain 0<α0 gives a inhomogeneous non-linearity. One considers the inter-critical regime, namely 1+\frac2(1-τ)N

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