Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics

Bongarti, Marcelo; Charoenphon, Sutthirut; Lasiecka, Irena

doi:10.1007/s00028-020-00654-2

Cited by 33 publications

(28 citation statements)

References 18 publications

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“…How small? This is an important question as argued in [4]. We will be able to show that some smallness will be only imposed at the lowest level of regularity, while higher derivatives can be large.…”

Section: Resultsmentioning

confidence: 77%

“…However, this presents several technical difficulties in the present scenario, even at the level of low frequencies (lower order terms). Hence, in this paper, we exploit another technique which, to the best of our knowledge, is new and makes a strong use of the fact that we only require initial data to be small in H. One of the advantages of such construction (for JMGT) was already exploited by the authors in [4] in allowing extension by density in the nonlinear environment. In this paper we discovered that it also allows to: a) prove global existence and exponential stability by the representation of the solution and twolevel stability of linear flows.…”

Section: Discussionmentioning

confidence: 99%

“…The parameter τ > 0 corresponds to time relaxation. For more about the physical interpretation of (1.1), its derivation and overall discussion see [6,19,10,11,42,13,20].This also includes an analysis of asymptotic behavior of solutions when the parameter of relaxation tends to zero [26,27,4,3] The issues of wellposedness and stability of solutions under homogeneous Dirichlet and Neumann boundary data were first addressed for both nonlinear and linearized (k = 0) dynamics around 2010 with the works of I. Lasiecka, R. Triggiani and B. Kaltenbacher [23,24,36,25]. For the analysis of the long-time dynamics of (1.1) for both linear and nonlinear cases, the function…”

Section: Pde Model and An Overviewmentioning

confidence: 99%

“…(5.9)Now we estimate the supremmum of the L 2 -norm of I 1 . For this we notice that a combination of Holder's Inequality with the Sobolev Embedding H 2 Γ 1 (Ω) ֒→ L4 (Ω) yieldsI 1 (t) 2 = (v t + w t )(v t − w t ) 2 ( v t 4 + w t 4 ) v t − w t 4 ( ∇v t 2 + ∇w t 2 ) ∇(v t − w t ) 2 β Ψ 1 − Ψ 2 X β , for each t. Then sup t∈[0,T ] I 1 (t) 2 β Ψ 1 − Ψ 2 X β .Next, for estimating the suppremum of the L 2 -norm of I 2 we notice that the sobolev embedding H 2 Γ 1 (Γ) ֒→ L ∞ (Ω) yields…”

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confidence: 94%

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Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

Bongarti¹,

Lasiecka²

2021

Preprint

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Boundary feedback stabilization of a critical, nonlinear Jordan-Moore-Gibson-Thompson (JMGT) equation is considered. JMGT arises in modeling of acoustic waves involved in medical/engineering treatments like lithotripsy, thermotherapy, sonochemistry, or any other procedures using High Intensity Focused Ultrasound (HIFU). It is a well-established and recently widely studied model for nonlinear acoustics (NLA): a third-order (in time) semilinear Partial Differential Equation (PDE) with the distinctive feature of predicting the propagation of ultrasound waves at finite speed due to heat phenomenon know as second sound which leads to the hyperbolic character of heat propagation. In practice, the JMGT dynamics is largely used for modeling the evolution of the acoustic velocity and, most importantly, the acoustic pressure as sound waves propagate through certain media. In this work, critical refers to (usual) case where mediadamping effects are non-existent or non-measurable and therefore cannot be relied upon for stabilization purposes.In this paper the issue of boundary stabilizability of originally unstable (JMGT) equation is resolved. Motivated by modeling aspects in HIFU technology, boundary feedback is supported only on a portion of the boundary, while the remaining part of the boundary is left free (available to control actions) . Since the boundary conditions imposed on the "free" part of the boundary fail to satisfy Lopatinski condition (unlike Dirichlet boundary conditions), the analysis of uniform stabilization from the boundary becomes very subtle and requires careful geometric considerations.

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Section: Resultsmentioning

confidence: 77%

Section: Discussionmentioning

confidence: 99%

Section: Pde Model and An Overviewmentioning

confidence: 99%

mentioning

confidence: 94%

See 2 more Smart Citations

Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

Bongarti¹,

Lasiecka²

2021

Preprint

Self Cite

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“…The results on the well-posedness of the Westervelt equation with an additional strong nonlinear damping and with L ∞ (Ω) varying coefficients have been obtained in [4,25]. We mention that this wave equation can also be rigorously recovered in the limit of a thirdorder nonlinear acoustic equation for vanishing thermal relaxation time; see the analysis in [3,16].…”

Section: Introductionmentioning

confidence: 90%

Local well-posedness of a coupled Westervelt-Pennes model of nonlinear ultrasonic heating

Nikolić¹,

Said‐Houari²

2021

Preprint

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High-Intensity Focused Ultrasound (HIFU) waves are known to induce localized heat to a targeted area during medical treatments. In turn, the rise in temperature influences their speed of propagation. This coupling affects the position of the focal region as well as the achieved pressure and temperature values. In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on the Westervelt wave equation coupled to the Pennes bioheat equation that captures this so-called thermal lensing effect. We prove that this quasi-linear model is wellposed locally in time and does not degenerate under a smallness assumption on the pressure data.

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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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Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics

Cited by 33 publications

References 18 publications

Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

Local well-posedness of a coupled Westervelt-Pennes model of nonlinear ultrasonic heating

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

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