The role of potential, Morawetz estimate and spacetime bound for quasilinear Schrödinger equations

2019/04/22 by Song, Xianfa
#35Q55 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1904.10342

Abstract

In this paper, we deal with the following Cauchy problem \ iu_t = Δu + 2uh'(|u|²)Δh(|u|²) + V(x)u, x∈ ℝ^N, t · gt;0
u(x,0) = u₀(x), x ∈ ℝ^N.. Here h(s) and V(x) are some real functions. We take the potential V(x)∈ L^q(ℝ^N)+L^∞(ℝ^N) as criterion of the blowup and global existence of the solution to (1.1). In some cases, we can classify it in the following sense: If V(x)∈ S(I), then the solution of (1.1) is always global existence for any u₀ satisfying 0q_c[L^q(ℝ^N)+L^∞(ℝ^N)], S(II)=\∪_q

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