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Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial case

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Abstract

In this paper we consider the inhomogeneous nonlinear Schrödinger (INLS) equation

$$\begin{aligned} i \partial _t u +\Delta u +|x|^{-b} |u|^{2\sigma }u = 0, \,\,\, x \in {\mathbb {R}}^N \end{aligned}$$

with \(N\ge 3\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma }\) satisfies \(0<s_c<1\). In a previous work, for radial initial data in \(\dot{H}^{s_c}\cap \dot{H}^1\), we prove the existence of blow-up solutions and also a lower bound for the blow-up rate. Here we extend these results to the non-radial case. We also prove an upper bound for the blow-up rate and a concentration result for general finite time blow-up solutions in \(H^1\).

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Notes

  1. This is the step where we explore the decaying factor in the nonlinearity instead of the radial assumption employed by Merle et al. [29, Theorem1.1].

  2. In [25, Theorem 1.2] the authors used in this part the radial Gagliardo-Nirenberg estimate

    $$\begin{aligned} \Vert f\Vert ^4_{L^4(|x|\ge R)}\le \frac{c}{R^2}\Vert \nabla f\Vert _{L^2(|x|\ge R)}\Vert f\Vert ^3_{L^2(|x|\ge R)}, \end{aligned}$$

    and hence, they need the radial restriction. Here, we use the decay of \(|x|^{-b}\) away from the origin to obtain the desired estimate in the general case.

  3. As in the proof of Theorems 1.1 and 1.2, this is the step where we use the decaying factor in the nonlinearity to replace the radial assumption.

  4. In the proof of [7, Lemma 4.4] does not require a radial assumption.

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Acknowledgements

LGF was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—CAPES, Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq and Fundação de Amparo a Pesquisa do Estado de Minas Gerais—Fapemig/Brazil.

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Authors and Affiliations

  1. Department of Mathematics, UFPI, Teresina, Brazil

    Mykael Cardoso

  2. Department of Mathematics, UFMG, Belo Horizonte, Brazil

    Luiz Gustavo Farah

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  1. Mykael Cardoso
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Cardoso, M., Farah, L.G. Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial case. Math. Z. 303, 63 (2023). https://doi.org/10.1007/s00209-023-03212-x

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