一类具有对数非线性项的四阶抛物型方程解的全局渐近性
Global Asymptotic Behavior of Solutions for a Class of Fourth-Order Logarithmic Nonlinear Parabolic Equations
摘要: 本文研究了一类具有对数非线性项的四阶抛物型方程的初值问题。通过运用势阱法及构造相应的能量泛函，证明了低初始能量条件下方程解的渐近性和爆破性，并给出了爆破时间的下界估计。
Abstract: In this paper, the initial value problem for a class of fourth-order parabolic equations with logarithmic nonlinear terms is studied. By applying the potential well method and constructing the corresponding energy general function, the asymptotic and bursting properties of the solutions are proved for low initial capacities, and lower bound estimates of the bursting time are given.
文章引用：彭迪, 石鹏. 一类具有对数非线性项的四阶抛物型方程解的全局渐近性[J]. 应用数学进展, 2020, 9(11): 2036-2045. https://doi.org/10.12677/AAM.2020.911236

参考文献

[1] Zangwill, A. (1996) Some Causes and a Consequence of Epitaxial Roughening. Journal of Crystal Growth, 163, 8-21. [Google Scholar] [CrossRef] 
[2] Schulze, T.P. and Kohn, R.V. (1999) A Geometric Model for Coarsening during Spiral-Model Growth of Thin Films. Physica D: Nonlinear Phenomena, 132, 520-542. [Google Scholar] [CrossRef] 
[3] King, B.B., Stein, O. and Winkler, M. (2003) A Fourth-Order Parabolic Equation Modeling Epitaxial Thin-Film Growth. Journal of Mathematical Analysis and Applications, 286, 459-490. [Google Scholar] [CrossRef] 
[4] Sattinger, D.H. (1968) On Global Solution of Nonlinear Hyperbolic Equations. Archive for Rational Mechanics and Analysis, 30, 148-172. [Google Scholar] [CrossRef] 
[5] Payne, L.E. and Sattinger, D.H. (1975) Saddle Points and Instability of Nonlinear Hyperbolic Equations. Israel Journal of Mathematics, 22, 273-303. [Google Scholar] [CrossRef] 
[6] Liu, Y.C. and Zhao, J.S. (2006) On Potential Wells and Applications to Semilinear Hyperbolic Equations and Parabolic Equations. Nonlinear Analysis: Theory, Methods & Applications, 64, 2665-2687. [Google Scholar] [CrossRef] 
[7] Chen, H. and Tian, S.Y. (2015) Initial Boundary Value Problem for a Class of Semilinear Pseudo-Parabolic Equations with Logarithmic Nonlinearity. Journal of Differential Equations, 258, 4424-4442. [Google Scholar] [CrossRef] 
[8] Liao, M.L. and Li, Q.W. (2020) A Class of Fourth-Order Parabolic Equations with Logarithmic Nonlinearity. Taiwanese Journal of Mathematics, 24, 975-1003. [Google Scholar] [CrossRef] 
[9] Han, Y.Z. (2018) A Class of Fourth-Order Parabolic Equation with Arbitrary Initial Energy. Nonlinear Analysis: Real World Applications, 43, 451-466. [Google Scholar] [CrossRef] 
[10] Ding, H. and Zhou, J. (2019) Global Existence and Blow-up for a Parabolic Problem of Kirchhoff Type with Logarithmic Nonlinearity. Applied Mathematics & Optimization, 1-57. [Google Scholar] [CrossRef] 
[11] Gazzola, F., Grunau, H.-C. and Sweers, G. (2010) Poluharmonic Boundary Value Problems. Spring, Berlin. [Google Scholar] [CrossRef] 

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