An application of L1 estimates for oscillating integrals to parabolic like semi-linear structurally damped σ-evolution models

Dao, Tuan Anh; Reissig, Michael

doi:10.1016/j.jmaa.2019.03.048

Cited by 34 publications

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“…The present paper is a continuation of our recent paper [4]. This paper is related to the case δ ∈ (0, σ 2 ).…”

mentioning

confidence: 80%

“…Abstract. The present paper is a continuation of our recent paper [4]. We will consider the following Cauchy problem for semi-linear structurally damped σ-evolution models:…”

Section: (Communicated By Irena Lasiecka)mentioning

confidence: 99%

“…On the one hand, compared to the regularity of the initial data we can see that a loss of regularity of the obtained Sobolev solutions to the semi-linear models appeared in [4]. This comes from the singular behavior of time-dependent coefficients as t → +0 in the (L m ∩ L q ) − L q and the L q − L q estimates of the solutions to the corresponding linear models with vanishing right-hand sides.…”

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confidence: 96%

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confidence: 99%

“…Moreover, taking account of the semi-linear models (1) and (2) we want to underline that the properties of the solutions change completely from (0, σ 2 ) to ( σ 2 , σ]. Here we propose to distinguish between "parabolic like models" in the case δ ∈ (0, σ 2 ) in the previous paper [4] (see, moreover, [1,3,15] for the structural damped wave equation in the case σ = 1) and "σ-evolution like models" in the case δ ∈ ( σ 2 , σ] in the present paper (in the case σ = 1, these so-called "hyperbolic like models" or "wave like models", have been studied in detail in [3,7]) according to expected decay estimates.…”

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confidence: 99%

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<inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> estimates for oscillating integrals and their applications to semi-linear models with <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-evolution like structural damping

Dao¹,

Reissig²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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The present paper is a continuation of our recent paper [4]. We will consider the following Cauchy problem for semi-linear structurally damped σ-evolution models:Our aim is to study two main models including σ-evolution models with structural damping δ ∈ ( σ 2 , σ) and those with visco-elastic damping δ = σ. Here the function f (u, ut) stands for power nonlinearities |u| p and |ut| p with a given number p > 1. We are interested in investigating the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on L q spaces by assuming additional L m regularity for the initial data, with q ∈ (1, ∞) and m ∈ [1, q).

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“…The present paper is a continuation of our recent paper [4]. This paper is related to the case δ ∈ (0, σ 2 ).…”

mentioning

confidence: 80%

“…Abstract. The present paper is a continuation of our recent paper [4]. We will consider the following Cauchy problem for semi-linear structurally damped σ-evolution models:…”

Section: (Communicated By Irena Lasiecka)mentioning

confidence: 99%

mentioning

confidence: 96%

“…

…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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<inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> estimates for oscillating integrals and their applications to semi-linear models with <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-evolution like structural damping

Dao¹,

Reissig²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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Critical exponent for semi‐linear structurally damped wave equation of derivative type

Dao

Fino

2020

Math Methods in App Sciences

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The main purpose of this paper is to study the following semi-linear structurally damped wave equation with nonlinearity of derivative type: u tt − Δu + (−Δ) ∕2 u t = |u t | p , u(0, x) = u 0 (x), u t (0, x) = u 1 (x), with > 0, n ≥ 1, ∈ (0, 2], and p > 1. In particular, we would like to prove the nonexistence of global weak solutions by using a new test function and suitable sign assumptions on the initial data in both the subcritical case and the critical case.

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Optimal large‐time estimates and singular limits for thermoelastic plate equations with the Fourier law

Chen

Ikehata

2023

Math Methods in App Sciences

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In this paper, we study asymptotic behaviors for classical thermoelastic plate equations with the Fourier law of heat conduction in the whole space , where we introduce a reduction methodology basing on third‐order (in time) differential equations and refined Fourier analysis. We derive optimal growth estimates when , bounded estimates when , and decay estimates when for the vertical displacement in the norm. Particularly, the new critical dimension for distinguishing the decisive role between the plate model and the Fourier law of heat conduction is discovered. Moreover, concerning the small thermal parameter in the temperature equation, we study the singular limit problem. We not only show global (in time) convergence of the vertical displacements between thermoelastic plates and structurally damped plates but also rigorously demonstrate a new second‐order profile of the solution. Our methodology can settle several closely related problems in thermoelasticity.

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An application of L1 estimates for oscillating integrals to parabolic like semi-linear structurally damped σ-evolution models

Cited by 34 publications

References 9 publications

Critical exponent for semi‐linear structurally damped wave equation of derivative type

Optimal large‐time estimates and singular limits for thermoelastic plate equations with the Fourier law

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