Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy

Lin, Qiang; Tian, Xueteng; Xu, Runzhang; Zhang, Meina

doi:10.3934/dcdss.2020160

Cited by 15 publications

(21 citation statements)

References 40 publications

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“…Indeed, such model is a fractional version of Kirchhoff model, which was introduced by Kirchhoff in [12] to consider elastic string vibrations. Recently, the fractional Kirchhoff problems have received more and more attention, some new existence results for fractional Kirchhoff problems were given, for example, in [4,18,34,28,32,33,24]. Actually, the study of problems like (1) has many significant applications, as explained by Caffarelli in [2], Laskin in [13] and Vázquez in [27].…”

Section: Introductionmentioning

confidence: 99%

Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

Xiang¹,

Hu²

2021

DCDS-S

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In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} u_{tt}+M([u]_{\alpha, 2}^2)(-\Delta)^{\alpha}u+(-\Delta)^{s}u_{t} = \int_{0}^{t}g(t-\tau)(-\Delta)^{\alpha}u(\tau)d\tau+\lambda|u|^{q -2}u, \end{align*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ M:[0, \infty)\rightarrow (0, \infty) $\end{document}</tex-math></inline-formula> is a nondecreasing and continuous function, <inline-formula><tex-math id="M2">\begin{document}$ [u]_{\alpha, 2} $\end{document}</tex-math></inline-formula> is the Gagliardo-seminorm of <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^\alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> are the fractional Laplace operators, <inline-formula><tex-math id="M6">\begin{document}$ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $\end{document}</tex-math></inline-formula> is a positive nonincreasing function and <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is a parameter. First, the local and global existence of solutions are obtained by using the Galerkin method. Then the global nonexistence of solutions is discussed via blow-up analysis. Our results generalize and improve the existing results in the literature.

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Section: Introductionmentioning

confidence: 99%

Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

Xiang¹,

Hu²

2021

DCDS-S

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“…where 2 < α < 2θ < p < 2 * < r. e global existence, behavior of solutions, and blow-up in time for (4) are obtained, under appropriate assumptions. In [18], the IBVP of Kirchhoff wave equation is considered. Under some sufficient conditions, the blow-up in finite time is shown by using a modified concavity method; for more details, see [19][20][21][22][23][24][25][26][27].…”

Section: Introduction and Brief History Of Fractional Integrodifferentiationmentioning

confidence: 99%

Blow-Up of Solutions for Wave Equation Involving the Fractional Laplacian with Nonlinear Source

Bidi

Beniani

Alnegga

et al. 2021

Mathematical Problems in Engineering

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“…An often indispensable tool is the elliptic theory [20,22,35] to establish the potential well theory. The potential well theory is a powerful tool for studying the dependence of the well-posedness of solution to the partial differential equations on the initial data, which can effectively deal with wave equations [28,45], the pseudo-parabolic equations with non-local source [37], polynomial source [41], singular potential [25], and the coupled parabolic systems [40].…”

mentioning

confidence: 99%

Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms

Pang¹,

Wang²,

Wu³

2021

DCDS-S

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We consider the well-posedness of solution of the initial boundary value problem to the fourth order wave equation with the strong and weak damping terms, and the logarithmic strain term, which was introduced to describe many complex physical processes. The local solution is obtained with the help of the Galerkin method and the contraction mapping principle. The global solution and the blowup solution in infinite time under sub-critical initial energy are also established, and then these results are extended in parallel to the critical initial energy. Finally, the infinite time blowup of solution is proved at the arbitrary positive initial energy.

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Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy

Abstract: In this paper, we study blow up and blow up time of solutions for initial boundary value problem of Kirchhoff-type wave equations involving the fractional Laplacian

Cited by 15 publications

References 40 publications

Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

Blow-Up of Solutions for Wave Equation Involving the Fractional Laplacian with Nonlinear Source

Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms

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