Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity

Liu, Xu; Zhou, Jun

doi:10.3934/era.2020032

Cited by 8 publications

(11 citation statements)

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“…where h (u) ∼ [u + 1] + − 1 describes restoring force due to the hangers and external forces including gravity and [u] + stands for its positive part. In other papers [5,6,19,20], this kind of problem was also investigated under the Navier boundary condition. More recently, Ferrero and Gazzola [8] considered the boundaries of a plate Ω = (0, π) × (−l, l) which represents the roadway of a suspension bridge.…”

Section: Introductionmentioning

confidence: 99%

Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge

Tran¹,

Pham²

2021

DCDS-S

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The paper deals with global existence and blow-up results for a class of fourth-order wave equations with nonlinear damping term and superlinear source term with the coefficient depends on space and time variable. In the case the weak solution is global, we give information on the decay rate of the solution. In the case the weak solution blows up in finite time, estimate the lower bound and upper bound of the lifespan of the blow-up solution, and also estimate the blow-up rate. Finally, if our problem contains an external vertical load term, a sufficient condition is also established to obtain the global existence and general decay rate of weak solutions.

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Section: Introductionmentioning

confidence: 99%

Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge

Tran¹,

Pham²

2021

DCDS-S

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“…Here α is the dimensionless order of fractional differentiation. If α = 1 then two above problem is called classical fourth order parabolic equations which can be studied in some interesting papers [17,16,25,24,34,33].…”

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Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation

Tuan¹

2021

DCDS-S

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In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source term. To overcome the intricacies of the proofs, we applied <inline-formula><tex-math id="M1">\begin{document}$ L^p-L^q $\end{document}</tex-math></inline-formula> estimate for biharmonic heat semigroup, Banach fixed point theory, some estimates for Mittag-Lefler functions and Wright functions, and also Sobolev embeddings. For the second result involving the Cahn-Hilliard equation with the Caputo-Fabrizio operator, we first show the local existence result. In addition, we first provide that the connections of the mild solution between the Cahn-Hilliard equation in the case <inline-formula><tex-math id="M2">\begin{document}$ 0<{\alpha}<1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ {\alpha} = 1 $\end{document}</tex-math></inline-formula>. This is the first result of investigating the Cahn-Hilliard equation with this type of derivative. The main key of the proof is based on complex evaluations involving exponential functions, and some embeddings between <inline-formula><tex-math id="M4">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> spaces and Hilbert scales spaces.

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“…In [14], Di considered the initial boundary value problem of the fourth order wave equation with an internal nonlinear source |u| ρ u, they proved the global existence and uniqueness of the regular solution and the weak solution respectively, and studied the explicit decay rate estimation of energy. Liu and Zhou [32] considered the local well-posedness of solutions to the initial boundary value problem for fourth-order plate equations with Hardy-Hénon potential and polynomial nonlinearity, and also studied the global existence and finite time blow-up results of solutions.…”

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confidence: 99%

A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms

Cheng

Chu

2021

era

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In this paper, we study a class of hyperbolic equations of the fourth order with strong damping and logarithmic source terms. Firstly, we prove the local existence of the weak solution by using the contraction mapping principle. Secondly, in the potential well framework, the global existence of weak solutions and the energy decay estimate are obtained. Finally, we give the blow up result of the solution at a finite time under the subcritical initial energy.

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Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity

Cited by 8 publications

References 50 publications

Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge

Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge

Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation

A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms

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