Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy

Xu, Runzhang; Wang, Xingchang; Yue, Yang

doi:10.1016/j.aml.2018.03.033

Cited by 54 publications

(35 citation statements)

References 6 publications

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“…Second, we prove G ⊂ Φ, where Φ is the set defined in (27). For any ϕ ∈ G, we need to show ϕ ∈ Φ, i.e.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

“…The homogeneous problem, i.e. σ = 0, was studied in [3,4,5,7,9,10,13,15,16,21,24,25,26,27,28,29]. Especially, for the Cauchy problem (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…the upper bound of the blow-up time) of the blowing-up solutions. Recently, Xu et al [27] and Han [9] extended the previous studies by considering the problem with general nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

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Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Zhou

2020

era

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This paper deals with the global existence and blow-up of solutions to a inhomogeneous pseudo-parabolic equation with initial value u 0 in the Sobolev space H 1 0 (Ω), where Ω ⊂ R n (n ≥ 1 is an integer) is a bounded domain. By using the mountain-pass level d (see (14)), the energy functional J (see (12)) and Nehari function I (see (13)), we decompose the space H 1 0 (Ω) into five parts, and in each part, we show the solutions exist globally or blow up in finite time. Furthermore, we study the decay rates for the global solutions and lifespan (i.e., the upper bound of blow-up time) of the blow-up solutions. Moreover, we give a blow-up result which does not depend on d. By using this theorem, we prove the solution can blow up at arbitrary energy level, i.e. for any M ∈ R, there exists u 0 ∈ H 1 0 (Ω) satisfying J(u 0) = M such that the corresponding solution blows up in finite time.

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“…Second, we prove G ⊂ Φ, where Φ is the set defined in (27). For any ϕ ∈ G, we need to show ϕ ∈ Φ, i.e.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

“…The homogeneous problem, i.e. σ = 0, was studied in [3,4,5,7,9,10,13,15,16,21,24,25,26,27,28,29]. Especially, for the Cauchy problem (i.e.…”

Section: Introductionmentioning

confidence: 99%

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Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Zhou

2020

era

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“…This method was first introduced by Sattinger [30] to investigate the global existence of solutions for nonlinear hyperbolic equations. Hence, it has been widely used and extended by many authors to study different kinds of evolution equations, we refer the reader to see [6,7,30,[34][35][36][38][39][40] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

Shang

2020

era

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In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks f 1 (uνt), f 2 (ut) and internal source |u| ρ u. Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks f 1 (uνt), f 2 (ut) have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we first obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function φ(t) (that depends on the behaviors of the functions f 1 (uνt), f 2 (ut) near the origin), nonlinear integral inequality and the Multiplier method.

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“…(1.2) There are many results for Eq. (1.2) such as the existence and uniqueness in [4], blow-up in [5][6][7][8], asymptotic behavior in [6,9], and so on. Using the integral representation and the semigroup, Cao et al [10] obtained the critical global existence exponent and the critical Fujita exponent for Eq.…”

Section: Introductionmentioning

confidence: 99%

Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level

Liu

Zhao

2018

Bound Value Probl

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In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation u t-u t-u-div(|∇u| 2q ∇u) = u p which was studied by Peng et al. (Appl. Math. Lett. 56:17-22, 2016), where the blow-up phenomena and the lifespan for the initial energy J(u 0) < 0 were obtained. We establish the finite time blow-up of the solution for the initial data at arbitrary energy level and the lifespan of the blow-up solution. Furthermore, as a product, we obtain the blow-up rate and refine the lifespan when J(u 0) < 0.

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Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy

Cited by 54 publications

References 6 publications

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level

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