Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Zhou, Jun

doi:10.3934/era.2020005

Cited by 12 publications

(9 citation statements)

References 28 publications

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“…We also refer to the recent papers [36,44,12,31,33,60,62,37,41,21,6] for the study of asymptotic behavior of semilinear pseudo-parabolic equations. If α = 1, m > 0, and 0 < σ = 1, Ji et al [29] considered the Cauchy problem of the following fractional pseudo-parabolic equation u t − m∆u t + (−∆) σ u = u p+1 , p > 0.…”

Section: 2mentioning

confidence: 99%

Semilinear Caputo time-fractional pseudo-parabolic equations

Tuan¹,

Au²,

Xu³

2021

Communications on Pure &Amp; Applied Analysis

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This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, we establish the local well-posedness theory including existence, uniqueness and regularity of the local solution, and the further local existence theory related to the finite time blow-up are also obtained for the problem with logarithmic nonlinearity. For the second problem with the source term satisfying the globally Lipschitz condition, we prove the global existence theorem.

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Section: 2mentioning

confidence: 99%

Semilinear Caputo time-fractional pseudo-parabolic equations

Tuan¹,

Au²,

Xu³

2021

Communications on Pure &Amp; Applied Analysis

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“…To be a little more precise, when

ω \geq 0

and

μ > - ω λ_{1}

, the local well‐posedness, global existence, and finite time blow‐up of solutions were studied in Ref. 2, 9 by using potential well theory, which was firstly introduced by Sattinger and Payne 11,12 , then developed by Liu and Zhao 13 (see also the references 14–21 for this method). Moreover, the lower bound of the blow‐up time for blow‐up solutions was investigated in Refs.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity

2021

Self Cite

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This paper investigates the infinite time blow‐up of solutions with arbitrary high initial energy to wave equations with weak damping term, strong damping term, and logarithmic nonlinearity. This problem has been studied previously with the assumptions that there is no strong damping term and the initial displacement and initial velocity have the same sign. However, from the physical point of view, it is obvious that the initial displacement and initial velocity may have different signs, and it is very necessary to consider the effects of the strong damping term. For example, the strong damping term indicates that the stress is not only proportional to the strain as with the Hooke law, but also proportional to the strain rate as in a linearized Kelvin–Voigt material. In this paper, by providing a completely different method from previous studies, we show that the solutions may blow up at infinity with arbitrary high initial energy when the model involves the strong damping term and the initial displacement and initial velocity may have different signs. Moreover, in this paper, we prove for the first time how to extend the solution over time (the whole half line) in studying the infinite time blow‐up phenomena for hyperbolic equations with logarithmic nonlinearity. These results fill in the gaps in previous studies on this type of models.

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“…leading to global existence, and we can even confirm this by knowledge of parabolic equation [2,10,16,18,26,29,28,32,33], we have not been able to find an effective way to prove it. Up to now, this question is still open.…”

mentioning

confidence: 93%

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

Tran¹,

Pham²

2021

DCDS-S

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<p style='text-indent:20px;'>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.</p>

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Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Cited by 12 publications

References 28 publications

Semilinear Caputo time-fractional pseudo-parabolic equations

Semilinear Caputo time-fractional pseudo-parabolic equations

Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity

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

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