Classification of blow-up and global existence of solutions to an initial Neumann problem

Guo, Bao‐Zhu; Zhang, Jingjing; Gao, Wenjie; Liao, Menglan

doi:10.1016/j.jde.2022.08.036

Cited by 12 publications

(10 citation statements)

References 30 publications

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“…They employed the extended concavity method to show some results about blow-up solutions. For more results, we referred the interested readers to [12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Note on a higher order pseudo‐parabolic equation with variable exponents

Liu

2023

Math Methods in App Sciences

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In this paper, we study a higher order pseudo‐parabolic equation involving ‐Laplacian with the Navier boundary condition. We use the energy method, the Sobolev embedding inequalities and the Galerkin's approximation to show the classification of singular solutions, including the existence and nonexistence of global, blow‐up, and extinction solutions. Extinction rates and time of extinction solutions, blow‐up time of blow‐up solutions, and decay estimates of global solutions are discussed.

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“…They employed the extended concavity method to show some results about blow-up solutions. For more results, we referred the interested readers to [12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Note on a higher order pseudo‐parabolic equation with variable exponents

Liu

2023

Math Methods in App Sciences

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“…Hence it is necessary and useful to extend this method to apply for PDEs with non-constant exponents. However, to our best knowledge, there are just a few results for PDEs with nonstandard growth conditions, see [13,27] for example. Also notice that as in the comments of the authors in [13], it is not a trivial extension even in case s = 0 because of the lack of homogeneity.…”

Section: Introductionmentioning

confidence: 99%

“…However, to our best knowledge, there are just a few results for PDEs with nonstandard growth conditions, see [13,27] for example. Also notice that as in the comments of the authors in [13], it is not a trivial extension even in case s = 0 because of the lack of homogeneity. When s = 0 the Kirchhoff function M then reduces constant function and (1.1) becomes the p(x)-Laplacian evolution equations…”

Section: Introductionmentioning

confidence: 99%

“…In [27] by developing the potential well method, we studied the existence and nonexistence of global weak solutions to the initial Dirichlet problem for (1.10) with α = 1 and f (u) = |u| m(x)−2 u. Afterward, by using a similar method the authors in [13] studied the same properties of weak solution to the initial Neumann problem for (1.10) with α = 1 and…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behavior of solutions for a new general class of parabolic Kirchhoff type equation with variable exponent sources

Quach

Nhan

2022

Preprint

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In this paper, we study the following problem:\begin{align*} \begin{cases} \displaystyle u_t -M\left(\int_{\Omega}\frac{\left|\nabla u\right|^{p(x)}}{p(x)}\diff x\right)\Delta_{p(x)}u = \left|u\right|^{m(x)-2}u,& (x,t)\in \Omega\times (0,T),\\ u(x,t)=0,&(x,t)\in \partial\Omega\times (0,T),\\ u(x,0)=u_0(x),&x\in \Omega, \end{cases}\end{align*}where $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, the functions $p,m:\bar{\Omega}\to\mathbb{R}$ and the Kirchhoff function $M:[0,\infty)\to [0,\infty)$ are specified below. We give a new class of general Kirchhoff function which covers both non-degenerate and degenerate cases, and investigate its effects on the existence and non-existence of global weak solutions to \eqref{1.1} in case the initial energy is subcritical. Moreover, we also give the decay estimate for the energy functional in the former case and an upper bound for the maximal existence time in the latter case. This is a continue working on Nhan et al., [Nonlinear Anal. Real World Appl., 56 (2020), 103155] and also an affirmative answer for the comments by Guo et al., [J. Differential Equation., 340 (2022), 45-82.] 2010 MSC: 35A01, 35B44, 35K55

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“…It is interesting that the blow‐up rates are determined by the maximal of the variable exponents. For more results on parabolic problems with nonstandard growth conditions, we refer the interested readers to the works 19–26 . And the recently related parabolic equations in the frame of potential well, we refer the interested readers to the works 27–29 …”

Section: Introductionmentioning

confidence: 99%

Energy classification in a nonstandard fourth‐order parabolic equation with a Navier boundary condition

Liu

Sun

Wang

2022

Math Methods in App Sciences

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This paper deals with a fourth-order parabolic equation involving variable exponents, subject to Navier boundary conditions. We firstly give an optimal classification on the initial energy for blow-up or global solutions, which is characterized by the sign of the Nehari energy and the difference between the initial energy and the depth of the potential well. Then, large time estimates of solutions are obtained. The results of the paper extend and complete the ones in "Applied Mathematics Letters 78(2018)141-146." KEYWORDS asymptotic behavior, blow-up, extinction, higher order parabolic equation, potential well method MSC CLASSIFICATION

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Classification of blow-up and global existence of solutions to an initial Neumann problem

Cited by 12 publications

References 30 publications

Note on a higher order pseudo‐parabolic equation with variable exponents

Note on a higher order pseudo‐parabolic equation with variable exponents

Asymptotic behavior of solutions for a new general class of parabolic Kirchhoff type equation with variable exponent sources

Energy classification in a nonstandard fourth‐order parabolic equation with a Navier boundary condition

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