Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS

We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in R N , i ∂ t u + Δ 2 u - | x | - b | u | 2 σ u = 0 , where σ > 0 and b > 0 . We first study the local well-posedness in H ˙ s c ∩ H ˙ 2 , for N ≥ 5 and 0 < s c < 2 , where s c = N 2 - 4 - b 2 σ . Next, we e...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2022-05, Vol.198 (1), p.1-29
Main Authors: Cardoso, Mykael, Guzmàn, Carlos M., Pastor, Ademir
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Well posed problems
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Summary:We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in R N , i ∂ t u + Δ 2 u - | x | - b | u | 2 σ u = 0 , where σ > 0 and b > 0 . We first study the local well-posedness in H ˙ s c ∩ H ˙ 2 , for N ≥ 5 and 0 < s c < 2 , where s c = N 2 - 4 - b 2 σ . Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in H ˙ s c ∩ H ˙ 2 with 0 ≤ s c < 2 . Finally, we study the phenomenon of L σ c -norm concentration for finite time blow up solutions with bounded H ˙ s c -norm, where σ c = 2 N σ 4 - b . Our main tool is the compact embedding of L ˙ p ∩ H ˙ 2 into a weighted L 2 σ + 2 space, which may be seen of independent interest.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-021-01667-w