Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case

Lai, Ning An; Takamura, Hiroyuki

doi:10.1016/j.na.2017.12.008

Cited by 63 publications

(59 citation statements)

References 32 publications

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“…Let us recall the definition of some multipliers related to our model, which have been introduced in [25], and some properties of them, that we will employ throughout the remaining sections.…”

Section: Iteration Framementioning

confidence: 99%

Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

Palmieri

Takamura

2019

Mediterr. J. Math.

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In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type, namely, in one equation a power nonlinearity and in the other a semilinear term of derivative type. The proof of the blow-up results is based on an iteration argument. As expected, due to the assumptions on the coefficients of the damping terms, we find as critical curve in the pq plane for the pair of exponents (p, q) in the nonlinear terms the same one found by Hidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we provide a different approach from the test function method applied by Ikeda-Sobajima-Wakasa to prove the blow-up of the solution on the critical curve, improving in some cases the upper bound estimate for the lifespan. More precisely, we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.

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Section: Iteration Framementioning

confidence: 99%

Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

Palmieri

Takamura

2019

Mediterr. J. Math.

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“…We also refer the reader to D'Abbicco [2] and D'Abbicco-Lucente-Reissig [5] for global existence results and determination of the critical exponent for the special case µ = 2, respectively. In the scattering case b(t) = (1 + t) −β with β > 1, Lai-Takamura [15] proved the blowup result for 1 < p < p S (N), and therefore, in this case the damping term can be ignored.…”

Section: Introductionmentioning

confidence: 95%

Sharp lifespan estimates of blowup solutions to semi-linear wave equations with time-dependent effective damping

Ikeda

Sobajima

Wakasugi

2019

J. Hyper. Differential Equations

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In this paper we consider the initial value problem for the semilinear wave equation with an effective dampingwhere N ∈ N, 0 < b ∈ C 1 ([0, ∞)), 1 < p ≤ 1 + 2 N and ε > 0 is a parameter describing the smallness of initial data. Here the coefficient b(t) of the damping term is assumed to be "effective". The interest is the behavior of lifespan of solutions in view of the asymptotic profile of b(t) as t → ∞. The simple cases b(t) = (1 + t) −β (β ∈ (−1, 1)) and the threshold case b(t) = 1+t in the sense of overdamping are discussed in Ikeda-Wakasugi [13] and Ikeda-Inui [9], respectively. In the present paper we discuss general damping terms with a certain assumption. The result of this paper is the sharp lifespan estimates of blowup solutions to (DW) including the typical case b(t) = (1 + t)(1 + log(1 + t)). The proof of upper bound of lifespan is a modification of the test function method given in [12] and the one of lower bound is based on the technique of scaling variables introduced in Gallay-Raugel [8] (for N = 1) and Wakasugi [24] (for N ≥ 2). (2010): Primary: 35L71. Mathematics Subject Classification

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“…Remark 2.3. As we have already mentioned the proof of Lemma 2.2 follows the approach from Section 3 in [18]. However, the same estimates can be proved by following the proof of Lemma 5.1 in [35], by working with a different functional in place of U 1 .…”

Section: Lemma 22mentioning

confidence: 76%

“…Now we can proceed with the second part of the proof, where we use a standard iteration argument (see for example [18,34] in the cae of a single equation or [1,25] in the case of a weakly coupled system). We will apply an iteration method based on lower bound estimates (19), (20), (29), (31) and the iteration frame (30), (32).…”

Section: Iteration Argumentmentioning

confidence: 99%

“…The remaining part of this paper is organized as follows: in Section 2 we recall a multiplier, that has been introduced in [18] in order to study the corresponding single semilinear equation, and its properties and we derive some lower bounds for certain functionals related to a local solution; then, in Section 3 we prove Theorem 1.2 by using the preparatory results from Section 2 and an iterative method. Finally, in Section 4 we prove the result in the critical case adapting the approach from [35,36] for a weakly coupled system.…”

Section: Introductionmentioning

confidence: 99%

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Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities

Palmieri

Takamura

2019

Nonlinear Analysis

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In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical case. In the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions. In the critical case we use the slicing method and a couple of auxiliary functions, recently introduced by Wakasa-Yordanov, to modify the definition of the functionals with the introduction of weight terms. In particular, we find as critical curve for the pair (p, q) of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities.

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Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case

Cited by 63 publications

References 32 publications

Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

Sharp lifespan estimates of blowup solutions to semi-linear wave equations with time-dependent effective damping

Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities

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