Global existence for the Jordan–Moore–Gibson–Thompson equation in Besov spaces

Said‐Houari, Belkacem

doi:10.1007/s00028-022-00788-5

Cited by 8 publications

(8 citation statements)

References 35 publications

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“…Summing up the estimates (32), ( 33), ( 34) and (35), we have the desired estimate (29). Another estimate (30) can be demonstrated by a parallel way. Next, we show the estimate (31), whose problematic part is the D 2 (t, ξ) because the lack of decay factor |ξ| in the derived estimate (34).…”

Section: Asymptotic Expansions Of Solution For Small Frequenciesmentioning

confidence: 96%

“…Concerning other related works of the JMGT equation, we refer interested readers to [30] for Besov spaces framework, [21] for inviscid limits with respect to the diffusivity of sound, [3] for singular limits with respect to the thermal relaxation, [26] for hereditary fluids, and [22] for the timefractional derivative models. Nevertheless, to the best of authors' knowledge, large-time asymptotic profiles for the solutions do not be deeply investigated, particularly, the second-order profiles (even for the linearized problem).…”

Section: Mathematical Researches On the Jmgt Equationmentioning

confidence: 99%

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Asymptotic behaviors for the Jordan-Moore-Gibson-Thompson equation in the viscous case

Chen¹,

Takeda²

2022

Preprint

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In this paper, we study large-time behaviors for a fundamental model in nonlinear acoustics, precisely, the viscous Jordan-Moore-Gibson-Thompson (JMGT) equation in the whole space R n . This model describes nonlinear acoustics in perfect gases under irrotational flow and equipping Cattaneo's law of heat conduction. By employing refined WKB analysis and Fourier analysis, we derive first-and second-order asymptotic profiles of solution to the Moore-Gibson-Thompson (MGT) equation as t ≫ 1, which illustrates novel optimal estimates for the solutions even subtracting its profiles. Concerning the nonlinear JMGT equation, via suggesting a new decomposition of nonlinear portion, we investigate the existence and large-time profiles of global (in time) small data Sobolev solutions with suitable regularity. These results help bridge a new connection between the JMGT equation and diffusion-waves as t ≫ 1.

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Section: Asymptotic Expansions Of Solution For Small Frequenciesmentioning

confidence: 96%

Section: Mathematical Researches On the Jmgt Equationmentioning

confidence: 99%

Asymptotic behaviors for the Jordan-Moore-Gibson-Thompson equation in the viscous case

Chen¹,

Takeda²

2022

Preprint

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“…Note that the remaining case n = 8 3 σ when η ∈ (3, ∞) does not include in (31), which will be discussed in Remark 4.2. The time-weighted Sobolev norm (36) is strongly motivated by the sharp (L 2 ∩ L 1 ) − L 2 type estimates for the linearized Cauchy problem (14) with η ∈ (1, ∞), precisely, Propositions 3.2-3.4 and Corollary 3.1.…”

Section: Global (In Time) Well-posedness For the Semilinear Cauchy Pr...mentioning

confidence: 99%

“…Then, with the same procedure as the above one, we also can demonstrate (40) provided that the exponent p satisfies (48). (31) holds, the following estimates for the linearized Cauchy problem (14) can be proved easily:…”

Section: Proof Of Theorem 21: Global (In Time) Existence Of Sobolev S...mentioning

confidence: 99%

“…Note that the critical exponent p crit (n, σ) proposed in (2) is not a generalization of the Fujita exponent p Fuj (n). Lastly, we also refer interested readers to another kind of important third-order (in time) PDEs, which is the so-called Jordan-Moore-Gibson-Thompson equation [23,29,3,24,31] in recent studies of nonlinear acoustics models in thermoviscous flows.…”

Section: Introductionmentioning

confidence: 99%

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Critical exponent and sharp lifespan estimates for semilinear third-order evolution equations

Chen¹

2023

Preprint

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We study semilinear third-order (in time) evolution equations with fractional Laplacian (−∆) σ and power nonlinearity |u| p , which was proposed by Bezerra-Carvalho-Santos [2] recently. In this manuscript, we obtain a new critical exponent p = pcrit(n, σ) := 1 + 6σ max{3n−4σ,0} for n 10 3 σ. Precisely, the global (in time) existence of small data Sobolev solutions is proved for the supercritical case p > pcrit(n, σ), and weak solutions blow up in finite time even for small data if 1 < p pcrit (n, σ). Furthermore, to more accurately describe the blow-up time, we derive new and sharp upper bound as well as lower bound estimates for the lifespan in the subcritical case and the critical case.

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Critical exponent and sharp lifespan estimates for semilinear third‐order evolution equations

Chen

2024

Math Methods in App Sciences

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We study semilinear third‐order (in time) evolution equations with fractional Laplacian and power nonlinearity , which was proposed by Bezerra–Carvalho–Santos (J. Evol. Equ. 2022) recently. In this manuscript, we obtain a new critical exponent for . Precisely, the global (in time) existence of small data Sobolev solutions is proved for the supercritical case , and energy solutions blow up in finite time even for small data if . Furthermore, to more accurately describe the blow‐up time, we derive new and sharp upper bound and lower bound estimates for the lifespan in the subcritical case and the critical case.

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Global existence for the Jordan–Moore–Gibson–Thompson equation in Besov spaces

Cited by 8 publications

References 35 publications

Asymptotic behaviors for the Jordan-Moore-Gibson-Thompson equation in the viscous case

Asymptotic behaviors for the Jordan-Moore-Gibson-Thompson equation in the viscous case

Critical exponent and sharp lifespan estimates for semilinear third-order evolution equations

Critical exponent and sharp lifespan estimates for semilinear third‐order evolution equations

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