@article{2802b058e6234a5ab45b621a0da12945,
title = "Focusing NLKG equation with singular potential",
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 utt - Δu + m2u = |x|-a|u|p-1u 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 = "Blow up, Critical energy, Global existence, Ground state",
author = "Vladimir Georgiev and Sandra Lucente",
note = "Funding Information: 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 Probabilit{\`a} 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 Probabilit{\'a} e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) Progetto 2017 Equazioni di tipo dispersivo e propriet{\`a} asintotiche. Funding Information: 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 ” Di-namiche Dispersive: Analisi di Fourier e Metodi Variazionali”, 2012; by INDAM, GNAMPA - Gruppo Nazionale per l{\textquoteright}Analisi Matematica, la Probabilit{\`a} 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{\textquoteright}Analisi Matematica, la Probabilit{\`a} e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) Progetto 2017 Equazioni di tipo dispersivo e propriet{\`a} asintotiche . ∗ Corresponding author: Sandra Lucente. Publisher Copyright: {\textcopyright} 2018 American Institute of Mathematical Sciences. All rights reserved.",
year = "2018",
month = jul,
doi = "10.3934/cpaa.2018068",
language = "English",
volume = "17",
pages = "1387--1406",
journal = "Communications on Pure and Applied Analysis",
issn = "1534-0392",
publisher = "American Institute of Mathematical Sciences",
number = "4",
}