Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

Nascimento, Wanderley Nunes do; Palmieri, Alessandro; Reissig, Michael

doi:10.1002/mana.201600069

Cited by 51 publications

(79 citation statements)

References 24 publications

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“…Let us remark that the condition

p_{prefixFuj} (n + \frac{μ_{1} - 1}{2} - \frac{\sqrt{δ}}{2}) \geq 2

is equivalent to

μ_{1} \leq 5 - 2 n + \sqrt{δ}

, while the inequality

\begin{matrix} true \\ right p_{Fuj} (() n + \frac{μ_{1} - 1}{2} - \frac{\sqrt{δ}}{2}) \leq \frac{n}{n - 2 σ} \end{matrix}

is equivalent to the restriction from below

(0false \frac{1}{σ} - 1) 2 n - 3 + \sqrt{δ} \leq μ_{1}

. Therefore, combining Theorem 2.4 in and Theorem , we have that

p_{prefixFuj} (n + \frac{μ_{1} - 1}{2} - \frac{\sqrt{δ}}{2})

is the critical exponent of , provided that the coefficients

μ_{1}, μ_{2}

and the exponent

σ \in [\frac{n}{4}, 1)

satisfy the conditions

\begin{matrix} true \\ right δ \geq 4 σ^{2} and ({, \begin{matrix} 2 (0true \frac{1}{σ} - 1) - 3 + \sqrt{δ} \leq μ_{1} \leq 3 + \sqrt{δ} & if σ \in [0true \frac{1}{4}, 0true \frac{1}{2}], \\ μ_{1} \leq 5 - 2 n + \sqrt{δ} & if σ \in (0true \frac{1}{2}, 1), \end{matrix}) \end{matrix}

…”

Section: Global Existence Of Small Data Solutionsmentioning

confidence: 78%

“…In Section we have seen how to improve the upper bound for the exponent p in the case

n > 2 σ

by using higher regularity of the data. However, our approach requires in this case a stronger condition on the coefficients μ 1 and μ 2 since in the statement of Theorem it is assumed that

δ \geq {(n + 2 σ - 1)}^{2}

, while in Theorem 2.2 of it is just required

δ \geq {(n + 1)}^{2}

.…”

Section: Global Existence Of Small Data Solutionsmentioning

confidence: 99%

“…For the proof of this last blow‐up result we refer to . Remark Let us underline that in the previous blow‐up result we can weaken the assumptions on data, requiring only

u_{0}, u_{1} \in L_{prefixloc}^{1}

such that holds.…”

Section: Introductionmentioning

confidence: 96%

“…The present work is a continuation of , in which global existence (in time) of small data solutions and blow‐up behavior of energy solutions are derived for the semi‐linear Cauchy problem, assuming

false(u_{0}, u_{1} false) \in {scriptD}^{1}

only and under certain conditions on the number

δ : = {(μ_{1} - 1)}^{2} - 4 μ_{2}^{2} .

…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, the main goal of this paper is to generalize some results of to the case in which

σ > 0

is different from 1, describing the σ‐dependent range of values for p and δ for which it is possible to prove a global (in time) existence and uniqueness result for the solutions to .…”

Section: Introductionmentioning

confidence: 99%

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Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Palmieri

Reissig

2018

Mathematische Nachrichten

Self Cite

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This paper is a continuation of our recent paper . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L2 scale with additional L1 regularity for the data. In order to deal with this L2 regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.

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“…Let us remark that the condition

p_{prefixFuj} (n + \frac{μ_{1} - 1}{2} - \frac{\sqrt{δ}}{2}) \geq 2

is equivalent to

μ_{1} \leq 5 - 2 n + \sqrt{δ}

, while the inequality

\begin{matrix} true \\ right p_{Fuj} (() n + \frac{μ_{1} - 1}{2} - \frac{\sqrt{δ}}{2}) \leq \frac{n}{n - 2 σ} \end{matrix}

is equivalent to the restriction from below

(0false \frac{1}{σ} - 1) 2 n - 3 + \sqrt{δ} \leq μ_{1}

. Therefore, combining Theorem 2.4 in and Theorem , we have that

p_{prefixFuj} (n + \frac{μ_{1} - 1}{2} - \frac{\sqrt{δ}}{2})

is the critical exponent of , provided that the coefficients

μ_{1}, μ_{2}

and the exponent

σ \in [\frac{n}{4}, 1)

satisfy the conditions

\begin{matrix} true \\ right δ \geq 4 σ^{2} and ({, \begin{matrix} 2 (0true \frac{1}{σ} - 1) - 3 + \sqrt{δ} \leq μ_{1} \leq 3 + \sqrt{δ} & if σ \in [0true \frac{1}{4}, 0true \frac{1}{2}], \\ μ_{1} \leq 5 - 2 n + \sqrt{δ} & if σ \in (0true \frac{1}{2}, 1), \end{matrix}) \end{matrix}

…”

Section: Global Existence Of Small Data Solutionsmentioning

confidence: 78%

“…In Section we have seen how to improve the upper bound for the exponent p in the case

n > 2 σ

by using higher regularity of the data. However, our approach requires in this case a stronger condition on the coefficients μ 1 and μ 2 since in the statement of Theorem it is assumed that

δ \geq {(n + 2 σ - 1)}^{2}

, while in Theorem 2.2 of it is just required

δ \geq {(n + 1)}^{2}

.…”

Section: Global Existence Of Small Data Solutionsmentioning

confidence: 99%

“…For the proof of this last blow‐up result we refer to . Remark Let us underline that in the previous blow‐up result we can weaken the assumptions on data, requiring only

u_{0}, u_{1} \in L_{prefixloc}^{1}

such that holds.…”

Section: Introductionmentioning

confidence: 96%

false(u_{0}, u_{1} false) \in {scriptD}^{1}

only and under certain conditions on the number

δ : = {(μ_{1} - 1)}^{2} - 4 μ_{2}^{2} .

…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, the main goal of this paper is to generalize some results of to the case in which

σ > 0

is different from 1, describing the σ‐dependent range of values for p and δ for which it is possible to prove a global (in time) existence and uniqueness result for the solutions to .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Palmieri

Reissig

2018

Mathematische Nachrichten

Self Cite

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show abstract

Energy behavior for Sobolev solutions to viscoelastic damped wave models with time‐dependent oscillating coefficient

2024

Mathematische Nachrichten

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In this work, we study the asymptotic behavior of the structurally damped wave equations arising from the viscoelastic mechanics. We are particularly interested in the complicated interaction of the time‐dependent oscillating coefficients on the Dirichlet Laplacian operator and the structurally damped terms. On the one hand, by the application of WKB analysis, we explore the asymptotic energy estimates of the wave equations influenced by four types of oscillating mechanisms. On the other hand, in order to prove the optimality of the energy estimates for the critical cases, typical coefficients and initial Cauchy data will be constructed to show the lower bound of the energy growth rate by the application of instability arguments.

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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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Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

Cited by 51 publications

References 24 publications

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation, II

Energy behavior for Sobolev solutions to viscoelastic damped wave models with time‐dependent oscillating coefficient

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

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