Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms

Sunagawa, Hideaki

doi:10.2969/jmsj/1149166781

Cited by 47 publications

(37 citation statements)

References 10 publications

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“…Note that analogous results have been obtained in [18] for the nonlinear KleinGordon equations and in [11], [13] for the nonlinear wave equations.…”

Section: Remark 4 When We Putsupporting

confidence: 73%

“…we can modify the above argument combining the idea of [4], [5] (see also Appendix of [18]) and show that the above theorem is still valid if N (1, iξ) Note that (6) is just what excludes u 3 , u 3 , uu 2 from all possible cubic nonlinear terms, but it is not a technical assumption because for these three nonlinearities we can find a class of initial data for which the solution has another kind of asymptotic profile than (5) (see [6] and [7] for the details).…”

Section: Remarkmentioning

confidence: 99%

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On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

Hayashi

Naumkin²,

Sunagawa

2008

SIAM J. Math. Anal.

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This talk is based on a joint work with Nakao Hayashi and Pavel Naumkin [8]. We consider the initial value problem for the nonlinear Shrödinger equation of the derivative type:whereand u is a complex-valued unknown function. We will occasionally write u x for ∂ x u, and u denotes the complex conjugate of u. The nonlinear term N(u, u x ) is a cubic homogeneous polynomial in (u, u, u x , u x ) with complex coefficients, and it satisfies so-called gauge invariance, that is,The aim of this talk is to present a structural condition on the nonlinear term N under which the corresponding forward Cauchy problem (1) has a disspative nature. To explain the motivation, let us begin with the simplest case where N is independent of u x , i.e., N = λ|u| 2 u with λ ∈ C. Then it is easy to see thatwhich suggests dissipativity if Im λ < 0. In fact, it is proved in [17] that the solution decays like O((t log t)x as t → +∞ if Im λ < 0 and u 0 is small enough. Since the non-trivial free solution (i.e., the solution to (1) for N ≡ 0, u 0 = 0) only decays like O(t −1/2 ), this gain of additional logarithmic time decay reflects a disspative character. Now we turn our attentions to the general gauge-invariant cubic nonlinear terms involving both u and u x . Note that we can not expect the conservation law like (3)

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“…Note that analogous results have been obtained in [18] for the nonlinear KleinGordon equations and in [11], [13] for the nonlinear wave equations.…”

Section: Remark 4 When We Putsupporting

confidence: 73%

Section: Remarkmentioning

confidence: 99%

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

Hayashi

Naumkin²,

Sunagawa

2008

SIAM J. Math. Anal.

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“…If λ ∈ C, another kind of long-range effect can be observed. Indeed, it is verified in [15] that the small data solution to (1.2) decays like O(t −1/2 (log t) −1/2 ) in L ∞ (R x ) as t → ∞ if Im λ < 0 (see also [17]). This gain of additional logarithmic time decay should be interpreted as another kind of long-range effect.…”

Section: Introductionmentioning

confidence: 92%

On Schrödinger systems with cubic dissipative nonlinearities of derivative type

Li¹,

Sunagawa²

2016

Nonlinearity

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“…Sunagawa studied in [28] the asymptotic behavior of small solutions of (4.1) under the condition that the initial data are regular, small and have a compact support. He showed that solutions have a more rapid time decay of order (t log (1 + t)) − 1 2 .…”

Section: Zampmentioning

confidence: 99%