Blow-up in damped abstract nonlinear equations

Esquivel-Avila, Jorge A.

doi:10.3934/era.2020020

Cited by 8 publications

(14 citation statements)

References 34 publications

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“…Two notable first articles on this subject are [30,29], see also [7,8,12,13,15,19,20,27,28], [31]- [34]. Related and interesting results have been reported in [2]- [5], [14], [16]- [18], [21,24,26,35]. The purpose of this work is to give sufficient conditions to get blow-up for the viscoelastic problem (P) with any positive value of the initial energy and improve the previous results.…”

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confidence: 87%

Nonexistence of global solutions for a class of viscoelastic wave equations

Esquivel-Avila¹

2021

DCDS-S

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<p style='text-indent:20px;'>We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.</p>

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confidence: 87%

Nonexistence of global solutions for a class of viscoelastic wave equations

Esquivel-Avila¹

2021

DCDS-S

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“…We want to know whether a slight change in the coefficients in the equations or boundary data, or even the equation itself, will lead to drastic changes in the solution. For a review of the nature of the structural stability, refer [13][14][15], Scott [16], Scott and Straughan [17], Payne et al [18][19][20][21] and some related papers [22][23][24]. The previous publications of structural stability usually study one fluid in a bounded domain.…”

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confidence: 99%

Structural stability for the Boussinesq equations interfacing with Darcy equations in a bounded domain

Zhang

Cheng

2021

Bound Value Probl

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A priori bounds were derived for the flow in a bounded domain for the viscous-porous interfacing fluids. We assumed that the viscous fluid was slow in $\Omega _{1}$ Ω 1 , which was governed by the Boussinesq equations. For a porous medium in $\Omega _{2}$ Ω 2 , we supposed that the flow satisfied the Darcy equations. With the aid of these a priori bounds we were able to demonstrate the result of the continuous dependence type for the Boussinesq coefficient λ. Following the method of a first-order differential inequality, we can further obtain the result that the solution depends continuously on the interface boundary coefficient α. These results showed that the structural stability is valid for the interfacing problem.

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“…Our main technical tool is the theory of potential wells (see e.g. [5,7,8,12,13,17,21,24,26,27,29,31]) with a slight modification, which plays an essential role in the proofs of main results.…”

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confidence: 99%