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July  2018, 17(4): 1387-1406. doi: 10.3934/cpaa.2018068

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

Received  January 2017 Revised  June 2017 Published  April 2018

Fund Project: 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

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: Ground state, critical energy, global existence, blow up.
Mathematics Subject Classification: Primary: 35L70; Secondary: 47J30, 35A01, 35B44.
Citation: Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
References:
[1]

V. Combet and F. Genoud, Classification of minimal mass blow-up solutions for an L2 critical inhomogeneous NLS, J-Evol. Eq., 16 (2016), 483-500. Google Scholar

[2]

Z. Gan and J. Zhang, Standing waves of the inhomogeneous Klein-Gordon equations with critical exponent, Acta Math. Sin. (Engl. Ser.), 22 (2006), 357-366. Google Scholar

[3]

Z. Gan and J. Zhang, Cross-constrained variational problem and the non-linear Klein-Gordon equations, Glasg. Math. J., 50 (2008), 467-481. Google Scholar

[4]

V. Georgiev and S. Lucente, Breaking Symmetry in focusing NLKG with potential, submittedGoogle Scholar

[5]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460. Errata arXiv: 1506.06248.Google Scholar

[6]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation, Transactions of the American Mathematical Society, 366 (2014), 5653-5669. Google Scholar

[7]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-upup for the energycritical focusing non-linear wave equation, Acta Mathematica, 201 (2008), 147-212. Google Scholar

[8]

E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, 2nd edition, AMS, 2001.Google Scholar

[9]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. Google Scholar

[10]

W. A. Strauss, Existence of solitary waves in higher dimension, Comm. Math. Phys., 55 (1977), 149-162. Google Scholar

[11]

J. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219. Google Scholar

[12]

E. Yanagida, Uniqueness of positive radial solutions of ∆u+g(r)u+h(r)up = 0 in ${{\mathbb{R}}^{n}}$, Arch. Rational Mech. Anal., 155 (1991), 257-274. Google Scholar

[13]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal. Ser. A: Theory Methods, 48 (2002), 191-207. Google Scholar

show all references

References:
[1]

V. Combet and F. Genoud, Classification of minimal mass blow-up solutions for an L2 critical inhomogeneous NLS, J-Evol. Eq., 16 (2016), 483-500. Google Scholar

[2]

Z. Gan and J. Zhang, Standing waves of the inhomogeneous Klein-Gordon equations with critical exponent, Acta Math. Sin. (Engl. Ser.), 22 (2006), 357-366. Google Scholar

[3]

Z. Gan and J. Zhang, Cross-constrained variational problem and the non-linear Klein-Gordon equations, Glasg. Math. J., 50 (2008), 467-481. Google Scholar

[4]

V. Georgiev and S. Lucente, Breaking Symmetry in focusing NLKG with potential, submittedGoogle Scholar

[5]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460. Errata arXiv: 1506.06248.Google Scholar

[6]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation, Transactions of the American Mathematical Society, 366 (2014), 5653-5669. Google Scholar

[7]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-upup for the energycritical focusing non-linear wave equation, Acta Mathematica, 201 (2008), 147-212. Google Scholar

[8]

E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, 2nd edition, AMS, 2001.Google Scholar

[9]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. Google Scholar

[10]

W. A. Strauss, Existence of solitary waves in higher dimension, Comm. Math. Phys., 55 (1977), 149-162. Google Scholar

[11]

J. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differ. Equ., 238 (2007), 201-219. Google Scholar

[12]

E. Yanagida, Uniqueness of positive radial solutions of ∆u+g(r)u+h(r)up = 0 in ${{\mathbb{R}}^{n}}$, Arch. Rational Mech. Anal., 155 (1991), 257-274. Google Scholar

[13]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Anal. Ser. A: Theory Methods, 48 (2002), 191-207. Google Scholar

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