TY - JOUR
T1 - Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions
AU - Hayashi, Nakao
AU - Naumkin, Pavel I.
N1 - Funding Information:
The authors would like to thank the referee for useful comments. This work of N.H. and P.I.N. was supported by KAKENHI (no. 19340030) and CONACYT, respectively.
PY - 2009/11/1
Y1 - 2009/11/1
N2 - We study asymptotics around the final states of solutions to the nonlinear Klein-Gordon equations with quadratic nonlinearities in two space dimensions (∂t2 - Δ + m2) u = λ u2, (t, x) ∈ R × R2, where λ ∈ C. We prove that if the final states u1+ ∈ Hfrac(q, q - 1)4 - frac(4, q) (R2) ∩ Hfrac(5, 2), 1 (R2) ∩ H12 (R2),u2+ ∈ Hfrac(q, q - 1)3 - frac(4, q) (R2) ∩ Hfrac(3, 2), 1 (R2) ∩ H11 (R2), and {norm of matrix} u1+ {norm of matrix}H12 + {norm of matrix} u2+ {norm of matrix}H11 is sufficiently small, where 4 < q ≤ ∞, then there exists a unique global solution u ∈ C ([T, ∞) ; L2 (R2)) to the nonlinear Klein-Gordon equations such that u (t) tends as t → ∞ in the L2 sense to the solution u0 (t) = u1+ cos (〈 i ∇ 〉m t) + (〈 i ∇ 〉m- 1 u2+) sin (〈 i ∇ 〉m t) of the free Klein-Gordon equation.
AB - We study asymptotics around the final states of solutions to the nonlinear Klein-Gordon equations with quadratic nonlinearities in two space dimensions (∂t2 - Δ + m2) u = λ u2, (t, x) ∈ R × R2, where λ ∈ C. We prove that if the final states u1+ ∈ Hfrac(q, q - 1)4 - frac(4, q) (R2) ∩ Hfrac(5, 2), 1 (R2) ∩ H12 (R2),u2+ ∈ Hfrac(q, q - 1)3 - frac(4, q) (R2) ∩ Hfrac(3, 2), 1 (R2) ∩ H11 (R2), and {norm of matrix} u1+ {norm of matrix}H12 + {norm of matrix} u2+ {norm of matrix}H11 is sufficiently small, where 4 < q ≤ ∞, then there exists a unique global solution u ∈ C ([T, ∞) ; L2 (R2)) to the nonlinear Klein-Gordon equations such that u (t) tends as t → ∞ in the L2 sense to the solution u0 (t) = u1+ cos (〈 i ∇ 〉m t) + (〈 i ∇ 〉m- 1 u2+) sin (〈 i ∇ 〉m t) of the free Klein-Gordon equation.
KW - Nonlinear Klein-Gordon equations
KW - Quadratic nonlinearity
KW - Two space dimensions
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U2 - 10.1016/j.na.2009.02.041
DO - 10.1016/j.na.2009.02.041
M3 - Article
AN - SCOPUS:67349192514
VL - 71
SP - 3826
EP - 3833
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
SN - 0362-546X
IS - 9
ER -