Large time behavior of solutions to semilinear systems of wave equations

Kubo, Hideo; Koyama, Koji; Sunagawa, Hideaki

doi:10.1007/s00208-006-0763-6

Cited by 19 publications

(39 citation statements)

References 13 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%

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%

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

Hayashi

Naumkin²,

Sunagawa

2008

SIAM J. Math. Anal.

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“…Suppose that θ 1 ≥ 0, i.e., if c 1 = c 2 , then p ≥ 2 ; otherwise p ≥ 3/2 (for the remaining case, we refer to [13]). As an unperturbed system, we choose…”

Section: )mentioning

confidence: 99%

Asymptotic behavior of solutions to semilinear systems of wave equations

Katayama¹,

Kubo²

2008

Indiana Univ. Math. J.

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“…The non-exitence of global in time solutions to (6) (which corresponds to (5) in the case µ 1 = µ 2 = 0 and ν 2 1 = ν 2 2 = 0) has been studied in [8,60], while the existence part has been proved in the three dimensional and radial case in [24]. Recently, in [15,Section 8] the upper bound for the lifespan has been derived.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, in order to guarantee the finiteness of U (z), we have to require the opposite inequality for z. Moreover, we are on the characteristict = z + R z, so from the inequality z ≤ R exp ( Eε 0 ) − pq−1 p+1we deduce for a suitable constantĒ > 0 the upper bound for the lifespanT (ε) ≤ exp Ē ε − pq−1 p+1 .Finally, switching the role between U and V (that is, working with lower bound estimates for V and applying the iteration frame (23)-(24) in the reverse order) we end up with the estimate T (ε) ≤ exp Eε − pq−1 q+1…”

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confidence: 99%

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

Palmieri

2019

Trends in Mathematics

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In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied. Keywords Blow-up, Glassey exponent, Nonlinearity of derivative type, Time-dependent and scale-invariant lower order terms, Integral representation formula, Upper bound estimates of the lifespan AMS Classification (2010) Primary: 35B44, 35L71; Secondary: 35B33, 35C15

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Large time behavior of solutions to semilinear systems of wave equations

Cited by 19 publications

References 13 publications

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

On the Schrödinger Equation with Dissipative Nonlinearities of Derivative Type

Asymptotic behavior of solutions to semilinear systems of wave equations

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

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