Asymptotic behaviour of solutions to some pseudoparabolic equations

Liu, Yan; Jiang, Weisheng; Huang, Fang‐Yang

doi:10.1016/j.aml.2011.07.012

Cited by 13 publications

(6 citation statements)

References 5 publications

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“…Problem describes varieties of important physical processes, such as the unidirectional propagation of nonlinear, dispersive, long waves, and the aggregation of population. There are many works about problem , such as the existence and uniqueness, asymptotic behavior, blow‐up,() and homogenization . Especially, in Xu and Su, the authors studied the global existence, blow‐up, and asymptotic behavior of solutions to problem by using the potential well method (see other works() for details of this method).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Lifespan for a semilinear pseudo‐parabolic equation

Zhou

2017

Math Methods in App Sciences

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This paper deals with the blow-up solution to the following semilinear pseudo-parabolic equationin a bounded domain Ω ⊂ R n , which was studied by Luo (Math Method Appl Sci 38 (12):2636-2641, 2015) with the following assumptions on p:and the lifespan for the initial energy J(u 0 ) < 0 is considered. This paper generalizes the above results on the following two aspects:1. a new blow-up condition is given, which holds for all p > 1; 2. a new lifespan is given, which holds for all p > 1 and possible J(u 0 ) ≥ 0.Moreover, as a byproduct, we refine the lifespan when J(u 0 ) < 0. KEYWORDS blow-up, lifespan, semilinear pseudo-parabolic equation p+1 p+1Math Meth Appl Sci. 2018;41:705-713. wileyonlinelibrary.com/journal/mma

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Lifespan for a semilinear pseudo‐parabolic equation

Zhou

2017

Math Methods in App Sciences

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“…Remark 2.3. In [5], we see that M = −∆(I − k∆) −1 is the infinitesimal generator of an analytic semigroup {e −Mt } t≥0 . This remark will be needed in the last section of this paper.…”

Section: Mittag-leffler Functionmentioning

confidence: 99%

Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

Phương

Tuấn²,

Kumar

et al. 2021

Math. Model. Nat. Phenom.

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In this paper, we investigate the initial boundary value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace of order $ 0<\nu\le1 $ and the nonlinear memory source term. For $ 0<\nu<1 $, the Problem will be considered on a bounded domain of $ \R^d $. By some Sobolev embeddings and the properties of Mittag-Lefler function, we will give some results on the existence and the uniqueness of mild solution for the Problem \eqref{Main-Equation} below. When $ \nu=1 $, we will introduce some $ L^p-L^q $ estimates, and base on them we derive the global existence of a mild solution in the whole space $ \R^d. $

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“…The asymptotic behavior of solutions to the initial value problem for the pseudo-parabolic equation was studied in a previous study. 27 The difference in our problem is the appearance of the condition (3); in fact, in many practical models, we cannot get accurate data either at the beginning or at the end (initial data or final data), which means we won't have the local conditions for our problem. Therefore, problems with non-local conditions help us to describe some phenomena more effectively, which is our motivation to investigate the problem with non-local conditions.…”

Section: Introductionmentioning

confidence: 99%

On time fractional pseudo‐parabolic equations with non‐local in time condition

Can

Kumar

Viet

et al. 2021

Math Methods in App Sciences

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The main objective of the paper is to study the non-local problem for a pseudoparabolic equation with fractional time and space. The derivative of time is understood in the sense of the time derivative of the Caputo fraction of the order 𝛼, 0 < 𝛼 < 1. The first result is an investigation of the existence and uniformity of the solution; the formula for mild solution and the regularity properties will be given. The proofs are based on a number of sophisticated techniques using the Sobolev embedding and also on the construction of the Mittag-Lefler operator. In the second part, we investigate the convergence of the mild solution for non-local problem to the solution of the local problem when two non-local parameters reach 0. Finally, we present some numerical examples to illustrate the proposed method.

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Asymptotic behaviour of solutions to some pseudoparabolic equations

Cited by 13 publications

References 5 publications

Lifespan for a semilinear pseudo‐parabolic equation

Lifespan for a semilinear pseudo‐parabolic equation

Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

On time fractional pseudo‐parabolic equations with non‐local in time condition

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