Global Solution and Blow-up for a Class of p-Laplacian Evolution Equations with Logarithmic Nonlinearity

Le, Cong Nhan; Le, Xuan Truong

doi:10.1007/s10440-017-0106-5

Cited by 59 publications

(37 citation statements)

References 10 publications

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“…In addition, Nhan and Truong in Le and Le 26 established the global existence, blow up, and decay of the solutions for p > 2 for the following problem: 11) as well as, Cao and Liu 27 proved global boundedness and blowing-up of problem (1.11) for 1 < p < 2. Motivated by above-mentioned papers, in this work, we investigate to prove the global existence, the growth, and the decay of solutions for problem (1.1), which was not previously studied, where we study the global existence of solutions by using potential well method of the p-Laplacian hyperbolic type equation with logarithmic nonlinearity and weak damping term, as well as the growth and the decay estimates of solutions for the problem are studied.…”

Section: P-laplacian Parabolic Equation With Logarithmic Source Termmentioning

confidence: 98%

“…Chen et al 25 proved the growth of the solutions for the following equation:

u_{t} - Δ u - Δ u_{t} = u \ln u .

In addition, Nhan and Truong in Le and Le 26 established the global existence, blow up, and decay of the solutions for p > 2 for the following problem:

u_{t} - div ((), {(||, \nabla u)}^{p - 2} \nabla u) - normalΔ u_{t} = {(||, u)}^{p - 2} u \ln (||, u),

as well as, Cao and Liu 27 proved global boundedness and blowing‐up of problem () for 1 < p < 2.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Qualitative analysis of solutions for the p‐Laplacian hyperbolic equation with logarithmic nonlinearity

Pışkın

Boulaaras

Irkıl

2020

Math Methods in App Sciences

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Section: P-laplacian Parabolic Equation With Logarithmic Source Termmentioning

confidence: 98%

“…Chen et al 25 proved the growth of the solutions for the following equation:

u_{t} - Δ u - Δ u_{t} = u \ln u .

In addition, Nhan and Truong in Le and Le 26 established the global existence, blow up, and decay of the solutions for p > 2 for the following problem:

u_{t} - div ((), {(||, \nabla u)}^{p - 2} \nabla u) - normalΔ u_{t} = {(||, u)}^{p - 2} u \ln (||, u),

as well as, Cao and Liu 27 proved global boundedness and blowing‐up of problem () for 1 < p < 2.…”

Section: Introductionmentioning

confidence: 98%

Qualitative analysis of solutions for the p‐Laplacian hyperbolic equation with logarithmic nonlinearity

Pışkın

Boulaaras

Irkıl

2020

Math Methods in App Sciences

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“…值得指出的是, 具对数非线性项的发展型方程解的整体存在和爆破性质已成为近年来研究的热点问题. 有兴趣的读者可以参见文献 [15][16][17]. 然而, 据我们所知, 对具对数非线性项的四阶方程解的整体存在和爆破性质的研究还没有任何结果, 这正是本文的主要研究方向.…”

Section: )unclassified

A class of thin-film equations with logarithmic nonlinearity

Han¹,

Gao²,

Shi³

2018

Sci. Sin.-Math.

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The main purpose of this paper is to study an initial-boundary value problem for a thin-film equation with logarithmic nonlinearity. Local existence of weak solutions is obtained by using Galerkin's approximation. By using the modified logarithmic Sobolev inequality and the potential well method, we obtain the existence of global solutions and solutions that blow up at infinity under some suitable conditions. The decay estimates of the global solutions are also derived.

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“…The blow-up solutions of parabolic p-Laplician equations have been studied by many authors (see, for instance, [1][2][3][4][5][6][7][8][9][10]). In this work, we research the blow-up solutions of the following parabolic p-Laplacian equations with a gradient source term:…”

Section: Introductionmentioning

confidence: 99%

Blow-up analysis for parabolic p-Laplacian equations with a gradient source term

Ding

2020

J Inequal Appl

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In this work, we deal with the blow-up solutions of the following parabolic p-Laplacian equations with a gradient source term: ⎧ ⎪ ⎨ ⎪ ⎩ (b(u)) t = ∇ • (|∇u| p-2 ∇u) + f (x, u, |∇u| 2 , t) in Ω × (0, t *), ∂u ∂n = 0 on ∂Ω × (0, t *), u(x, 0) = u 0 (x) ≥ 0 i n Ω, where p > 2, the spatial domain Ω ⊂ R N (N ≥ 2) is bounded, and the boundary ∂Ω is smooth. Our research relies on the creation of some suitable auxiliary functions and the use of the differential inequality techniques and parabolic maximum principles. We give sufficient conditions to ensure that the solution blows up at a finite time t *. The upper bounds of the blow-up time t * and the upper estimates of the blow-up rate are also obtained.

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Global Solution and Blow-up for a Class of p-Laplacian Evolution Equations with Logarithmic Nonlinearity

Cited by 59 publications

References 10 publications

Qualitative analysis of solutions for the p‐Laplacian hyperbolic equation with logarithmic nonlinearity

Qualitative analysis of solutions for the p‐Laplacian hyperbolic equation with logarithmic nonlinearity

A class of thin-film equations with logarithmic nonlinearity

Blow-up analysis for parabolic p-Laplacian equations with a gradient source term

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