Focusing nlkg equation with singular potential

Vladimir Georgiev; Sandra Lucente

doi:10.3934/cpaa.2018068

Article Contents

2018, Volume 17, Issue 4: 1387-1406. Doi: 10.3934/cpaa.2018068

This issue Previous Article On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion Next Article Asymptotics for the modified witham equation

Focusing nlkg equation with singular potential

Vladimir Georgiev^1,2, , and
Sandra Lucente^3,

1.
Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5 Pisa, 56127 Italy
2.
Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
3.
Department of Mathematics, University of Bari, Via E. Orabona 4 I-70125 Bari, Italy

^* Corresponding author: Sandra Lucente
^* Corresponding author: Sandra Lucente

Received: December 31, 2016

Revised: May 31, 2017

Published: April 2018

The first author was supported by University of Pisa, project no. PRA-2016-41"Fenomeni singolari in problemi deterministici e stocastici ed applicazioni"; by the Contract FIRB" Dinamiche Dispersive: Analisi di Fourier e Metodi Variazionali", 2012; by INDAM, GNAMPA -Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni; by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences; by Top Global University Project, Waseda University. The second author was supported in part by Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) Progetto 2017 Equazioni di tipo dispersivo e proprietà asintotiche.

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Abstract

We study the dynamics for the focusing nonlinear Klein Gordon equation with a positive, singular, radial potential and initial data in energy space. More precisely, we deal with

$u_{tt}-Δ u+m^2 u=|x|^{-a}|u|^{p-1}u$

with $0 < a < 2$. In dimension $d≥3$, we establish the existence and uniqueness of the ground state solution that enables us to define a threshold size for the initial data that separates global existence and blow-up. We find a critical exponent depending on $a$. We establish a global existence result for subcritical exponents and subcritical energy data. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary sets.

Keywords:

Mathematics Subject Classification: Primary: 35L70; Secondary: 47J30, 35A01, 35B44.

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References

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