Asymptotic Behavior of Solutions for a Fourth Order Parabolic Equation with Gradient Nonlinearity via the Galerkin Method

Asymptotic Behavior of Solutions for a Fourth Order Parabolic Equation with Gradient Nonlinearity via the Galerkin Method
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伽辽金法梯度非线性四阶抛物型方程解的渐近行为

DOI：
10.1007/978-3-030-73363-6_12
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发表时间：
2021
期刊：
Geometric Properties for Parabolic and Elliptic PDE's
影响因子：
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通讯作者：
Okabe Shinya
Okabe Shinya
中科院分区：
--
文献类型：
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作者：
Miyake Nobuhito;Okabe Shinya

文献摘要

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本文考虑一类具有梯度非线性项的四阶抛物型方程的初边值问题。该问题被视为能量泛函的L2-梯度流，该能量泛函从下到上是无界的。我们首先通过Galerkin方法证明了该问题解的存在唯一性。此外，结合势阱方法和Galerkin方法，我们研究了该问题的整体时间解的渐近性态。
In this paper we consider the initial-boundary value problem for a fourth order parabolic equation with gradient nonlinearity. The problem is regarded as theL2-gradient flow for an energy functional which is unbounded from below. We first prove the existence and the uniqueness of solutions to the problem via the Galerkin method. Moreover, combining the potential well method with the Galerkin method, we study the asymptotic behavior of global-in-time solutions to the problem.