Some sharp results about the global existence and blowup of solutions to a class of coupled pseudo-parabolic equations

Tran, Quang-Minh; Vu, Thi-Thi

doi:10.1016/j.jmaa.2021.125719

Cited by 11 publications

(2 citation statements)

References 27 publications

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“…Remark 2.4. We also note that Assumption 2.2 contains a large class of nonlinear source, which considered in the previous paper [5,6,[11][12][13]. Indeed, by direct calculation, it is easy to verify that the following nonlinear sources…”

Section: Notation and Primary Resultsmentioning

confidence: 76%

Classification of Blow-up and Global Existence of Solutions to a System of Petrovsky Equations

Truong

2023

Electron. J. Appl. Math.

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In this paper, we investigate global existence, uniform decay, and blow-up of solutions for a class of system of Petrovsky equations containing nonlinear damping and sources. By introducing a family of potential wells, we not only obtain the invariant sets and vacuum isolating of solutions but also give some threshold results of global existence and nonexistence of solutions. Furthermore, by using energy techniques, we also establish certain qualitative estimates for solution.

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Section: Notation and Primary Resultsmentioning

confidence: 76%

Classification of Blow-up and Global Existence of Solutions to a System of Petrovsky Equations

Truong

2023

Electron. J. Appl. Math.

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“…We will prove that in Section 5.2 and introduce a new method to fix the mistake. We note more that, in many mathematical literatures related to parabolic or pseudo-parabolic equations, many efforts with using a variety of methods have been devoted to the study of blow-up properties if the solutions blow up, see [6,15,18] and the references cited therein. In [6], Li et al used a differential inequality technique to derive the lower bound for the blow-up time when the blow-up occurs.…”

Section: Introductionmentioning

confidence: 99%

Some qualitative properties of weak solution for pseudo‐parabolic equation with viscoelastic term and Robin boundary conditions

Ngo,

Dao,

Freitas

2023

Mathematische Nachrichten

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In this paper, we consider the initial boundary value problem of the generalized pseudo‐parabolic equation containing viscoelastic terms and associated with Robin conditions. We establish first the local existence of solutions by the standard Galerkin method. Then, we prove blow‐up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow‐up rate for the weak solution by finding the upper bound and the lower bound for the blow‐up times and the upper bound and the lower bound for the blow‐up rate. For negative energy, we introduce a new method to prove blow‐up results with a sharper estimate for the upper bound for the blow‐up times. Finally, we prove both the global existence of the solution and the general decay of the energy functions under some restrictions on the initial data.

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Global existence and blowup of solutions for nonlinear Love equation

Dung,

Tran Vu

2024

Math Methods in App Sciences

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In this paper, we are concerned with the nonlinear Love equation, which describes some sort of “propagation process” in physics. Firstly, we establish the local existence and uniqueness of the solution by using the contraction mapping principle and the regularity theory for the elliptic problem. Secondly, we construct a family of potential wells to obtain a threshold for the existence of global solutions and blowup in a finite time of the solution in both cases with subcritical and critical initial energy. Additionally, we provide the decay rate of the global solution and estimate the lifespan of a blow‐up solution. Moreover, at the supercritical initial energy level, we provide a sufficient condition for initial data leading to blow‐up results. This paper, along with the previous work of A. N. Carvalho, J. Cholewa, Local well‐posedness, asymptotic behavior, and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time, Trans. Amer. Math. Soc. 361 (5) (2009) 2567‐2586, provides us with a comprehensive and systematic study of the dynamic behavior of the solution to the nonlinear Love equation.

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Some sharp results about the global existence and blowup of solutions to a class of coupled pseudo-parabolic equations

Cited by 11 publications

References 27 publications

Classification of Blow-up and Global Existence of Solutions to a System of Petrovsky Equations

Classification of Blow-up and Global Existence of Solutions to a System of Petrovsky Equations

Some qualitative properties of weak solution for pseudo‐parabolic equation with viscoelastic term and Robin boundary conditions

Global existence and blowup of solutions for nonlinear Love equation

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