A note on blow-up of solution for a class of semilinear pseudo-parabolic equations

Liu, Wenjun; Yu, Jiangyong

doi:10.1016/j.jfa.2018.01.005

Cited by 31 publications

(19 citation statements)

References 4 publications

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“…(1) If J(φ) < 0, then we can easily get from the definitions of J and I in (12) and (13) respectively that I(φ) < 0. So we have φ ∈ W if J(φ) < 0.…”

Section: Notations and 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%

“…In [28], Yang et al proved that for p > p c , there is a secondary critical exponent α c = 2/(p − 1) such that the solution blows up in finite time for u 0 behaving like |x| −α at |x| → ∞ if α = (0, α c ); and there are global solutions for for u 0 behaving like |x| −α at |x| → ∞ if α = (α c , n). For the zero Dirichlet boundary problem in a bounded domain Ω, in [13,25,26], the authors studied the properties of global existence and blow-up by potential well method (which was firstly introduced by Sattinger [19] and Payne and Sattinger [18], then developed by Liu and Zhao in [14]), and they showed the global existence, blowup and asymptotic behavior of solutions with initial energy at subcritical, critical and supercritical energy level. The results of [13,25,26] were extended by Luo [15] and Xu and Zhou [24] by studying the lifespan (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 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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“…(1) If J(φ) < 0, then we can easily get from the definitions of J and I in (12) and (13) respectively that I(φ) < 0. So we have φ ∈ W if J(φ) < 0.…”

Section: Notations and 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%

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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“…Obviously, (38) also holds for . Applying Schauder's estimate 39,40 for linear parabolic equation to (33), we have…”

Section: Reconstruction By Optimizationmentioning

confidence: 99%

A coefficient identification problem for a mathematical model related to ductal carcinoma in situ

Chen

et al. 2019

Stud Appl Math

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We consider a coefficient identification problem for a mathematical model with free boundary related to ductal carcinoma in situ (DCIS). This inverse problem aims to determine the nutrient consumption rate from additional measurement data at a boundary point. We first obtain a global‐in‐time uniqueness of our inverse problem. Then based on the optimization method, we present a regularization algorithm to recover the nutrient consumption rate. Finally, our numerical experiment shows the effectiveness of the proposed numerical method.

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“…A powerful technique for treating problem (1.2) is the so called "potential well method", which was established by Sattinger [22], Payne and Sattinger [21], and then improved by Liu and Zhao [17] by introducing a family of potential wells. Recently, there are some interesting results about the global existence and blow-up of solutions for problem (1.2) in [3], in which Chen and Tian proved global existence, blow-up at +∞, the behavior of vacuum isolation and asymptotic behavior of solutions with initial energy J(u 0 ) ≤ d. For other related works, we refer the readers to [2,11,6,15] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Global Existence, Exponential Decay and Blow-Up of Solutions for a Class of Fractional Pseudo-Parabolic Equations with Logarithmic Nonlinearity

Liu¹,

Yu²,

Li³

2018

Preprint

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A note on blow-up of solution for a class of semilinear pseudo-parabolic equations

Cited by 31 publications

References 4 publications

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

A coefficient identification problem for a mathematical model related to ductal carcinoma in situ

Global Existence, Exponential Decay and Blow-Up of Solutions for a Class of Fractional Pseudo-Parabolic Equations with Logarithmic Nonlinearity

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