Abstract
We study the dynamics for the focusing nonlinear Klein-Gordon equation, utt - Δu + m2u = V (x)|u|p-1u, with positive radial potential V and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo-Nirenberg inequality gives a critical exponent depending on V. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. 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 domains.
| Original language | English |
|---|---|
| Pages (from-to) | 755-788 |
| Number of pages | 34 |
| Journal | Journal of Hyperbolic Differential Equations |
| Volume | 15 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2018 Dec 1 |
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Keywords
- critical energy
- global existence/blow up
- Ground state
ASJC Scopus subject areas
- Analysis
- Mathematics(all)
Cite this
Breaking symmetry in focusing nonlinear Klein-Gordon equations with potential. / Gueorguiev, Vladimir Simeonov; Lucente, Sandra.
In: Journal of Hyperbolic Differential Equations, Vol. 15, No. 4, 01.12.2018, p. 755-788.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Breaking symmetry in focusing nonlinear Klein-Gordon equations with potential
AU - Gueorguiev, Vladimir Simeonov
AU - Lucente, Sandra
PY - 2018/12/1
Y1 - 2018/12/1
N2 - We study the dynamics for the focusing nonlinear Klein-Gordon equation, utt - Δu + m2u = V (x)|u|p-1u, with positive radial potential V and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo-Nirenberg inequality gives a critical exponent depending on V. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. 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 domains.
AB - We study the dynamics for the focusing nonlinear Klein-Gordon equation, utt - Δu + m2u = V (x)|u|p-1u, with positive radial potential V and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo-Nirenberg inequality gives a critical exponent depending on V. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. 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 domains.
KW - critical energy
KW - global existence/blow up
KW - Ground state
UR - http://www.scopus.com/inward/record.url?scp=85060109504&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85060109504&partnerID=8YFLogxK
U2 - 10.1142/S0219891618500248
DO - 10.1142/S0219891618500248
M3 - Article
VL - 15
SP - 755
EP - 788
JO - Journal of Hyperbolic Differential Equations
JF - Journal of Hyperbolic Differential Equations
SN - 0219-8916
IS - 4
ER -