Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent

Fu, Yongqiang; Xiang, Mingqi

doi:10.1080/00036811.2015.1022153

Cited by 21 publications

(12 citation statements)

References 27 publications

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“…Like the name in [27], we refer to (1) as the mixed pseudo-parabolic Kirchhoff equation, which with the combination of M (•) and p-Laplacian, can be used to describe the motion of a non-stationary fluid or gas in the nonhomogeneous and anisotropic medium [11], the growth and movement of biological species [13]... Especially, if p varies according to (x, t), then this type of problem can be applied to electrorheological fluids, nonlinear elastic and image restoration [34,35,25,36,9].…”

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confidence: 99%

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

Cao

Zhao

2021

era

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<p style='text-indent:20px;'>In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy <inline-formula><tex-math id="M1">\begin{document}$ J(u_0)\leq d $\end{document}</tex-math></inline-formula>. When the initial energy <inline-formula><tex-math id="M2">\begin{document}$ J(u_0)>d $\end{document}</tex-math></inline-formula>, we find another criterion for the vanishing solution and blow-up solution. Our interest also lies in the discussion of the exponential decay rate of the global solution and life span of the blow-up solution.</p>

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confidence: 99%

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

Cao

Zhao

2021

era

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“…Problem (1.3) can also be used to describe the motion of a nonstationary fluid or gas in a nonhomogeneous and anisotropic medium, and the nonlocal term M appearing in (1.3) can describe a possible change in the global state of the fluid or gas caused by its motion in the considered medium [6]. When f (x, t, u) ≡ f (x) and 0 < m ≤ M (s) ≤ M 0 for all s ≥ 0, Chipot et al investigated the existence, uniqueness and asymptotic behavior of solutions to (1.3) for both p = 2 and general p > 1 (see [2,3]).…”

Section: Copyright C 2019 the Author(s) Published By Vgtu Pressmentioning

confidence: 99%

“…Inspired by some ideas from [6,10,13,21,24], we shall consider the global existence and finite time blow-up of solutions to problem (1.1) for general p > 1, by combining the modified potential well method with the classical Galerkin's approximation and energy estimates. It is noteworthy that the results obtained here are not trivial generalization of that of the case p = 2 in [10].…”

Section: Copyright C 2019 the Author(s) Published By Vgtu Pressmentioning

confidence: 99%

Global Existence and Finite Time Blow-Up of Solutions to a Nonlocal P-Laplace Equation

Han

2019

Mathematical Modelling and Analysis

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In this paper a class of nonlocal diffusion equations associated with a p-Laplace operator, usually referred to as p-Kirchhoff equations, are studied. By applying Galerkin’s approximation and the modified potential well method, we obtain a threshold result for the solutions to exist globally or to blow up in finite time for subcritical and critical initial energy. The decay rate of the L 2 norm is also obtained for global solutions. When the initial energy is supercritical, an abstract criterion is given for the solutions to exist globally or to blow up in finite time, in terms of two variational numbers. These generalize some recent results obtained in [Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(9):3283–3297, 2018].

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“…Pulkina et al [15], and in the references therein. Besides, there are many works which focused on this topic; see, e.g., [16][17][18][19][20][21][22][23][24][25][26][27].…”

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confidence: 99%

On a Kirchhoff diffusion equation with integral condition

Nam¹,

Băleanu

Luc

et al. 2020

Adv Differ Equ

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This paper is devoted to Kirchhoff-type parabolic problem with nonlocal integral condition. Our problem has many applications in modeling physical and biological phenomena. The first part of our paper concerns the local existence of the mild solution in Hilbert scales. Our results can be studied into two cases: homogeneous case and inhomogeneous case. In order to overcome difficulties, we applied Banach fixed point theorem and some new techniques on Sobolev spaces. The second part of the paper is to derive the ill-posedness of the mild solution in the sense of Hadamard.

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Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent

Cited by 21 publications

References 27 publications

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

Global Existence and Finite Time Blow-Up of Solutions to a Nonlocal P-Laplace Equation

On a Kirchhoff diffusion equation with integral condition

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