TY - JOUR

T1 - The initial value problem for the quadratic nonlinear Klein-Gordon equation

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

PY - 2010

Y1 - 2010

N2 - We study the initial value problem for the quadratic nonlinear Klein-Gordon equation Lu = 〈i∂x〉-1 u-2, (t,x) ∈ R × R, u(0,x ) = u0(x), x ∈ R, where L = ∂t + i〈i∂x 〉 and 〈i∂ x〉 = 1 - ∂2 x. Using the Shatah normal forms method, we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data which was assumed in the previous works.

AB - We study the initial value problem for the quadratic nonlinear Klein-Gordon equation Lu = 〈i∂x〉-1 u-2, (t,x) ∈ R × R, u(0,x ) = u0(x), x ∈ R, where L = ∂t + i〈i∂x 〉 and 〈i∂ x〉 = 1 - ∂2 x. Using the Shatah normal forms method, we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data which was assumed in the previous works.

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U2 - 10.1155/2010/504324

DO - 10.1155/2010/504324

M3 - Article

AN - SCOPUS:84868380439

SN - 1687-9120

JO - Advances in Mathematical Physics

JF - Advances in Mathematical Physics

M1 - 504324

ER -