A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale‐invariant lower order terms

Palmieri, Alessandro

doi:10.1002/mma.6412

Cited by 23 publications

(16 citation statements)

References 45 publications

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“…This peculiarity of a "parabolic-like" behavior for large values of δ and of "wave-like" behavior for small values of δ has been showed also for the corresponding weakly coupled system (cf. [3,37]).…”

Section: Introductionmentioning

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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Section: Introductionmentioning

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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“…Furthermore, in the critical case, where we consider the not-damped case as in Section 9 of [16], it is interesting to compare how our different approach leads to different upper bound estimates for the lifespan and in some cases to an improvement of these estimates. We refer to [15] and to [14,16,36,33] for further details on this revised test function method based on a family of self similar solutions of the adjoint linear equation involving Gauss hypergeometric functions, for the study of semilinear heat, Schrödinger and damped wave equations and for the treatment of semilinear and scale-invariant model with time-dependent coefficients, respectively.…”

Section: Introductionmentioning

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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“…In these papers, the time -dependent multiplier is bounded by positive constants from above and from below and it is used to study semilinear damped wave models with time -dependent coefficients for the damping terms in the scattering producing case. On the other hand, the case of unbounded time -dependent multipliers is considered for semilinear wave models with scaleinvariant damping and mass terms in [24,22,35,31,36].…”

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confidence: 99%

A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case

Chen¹,

Palmieri²

2021

EECT

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In this paper, we study the blow-up of solutions to the semilinear Moore-Gibson-Thompson (MGT) equation with nonlinearity of derivative type |ut| p in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow-up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent p for the nonlinear term satisfies 1 < p (n + 1)/(n − 1) for n 2 and p > 1 for n = 1. In particular, we find the same blow-up range for p as in the corresponding semilinear wave equation with nonlinearity of derivative type.

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A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale‐invariant lower order terms

Cited by 23 publications

References 45 publications

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

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

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

A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case

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