Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity

Yuxuan Chen; Yuxuan Chen

doi:10.3934/cam.2023033

Communications in Analysis and Mechanics

2023, Volume 15, Issue 4: 658-694. doi: 10.3934/cam.2023033

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Research article

Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity

Yuxuan Chen ^,

School of Mathematical Sciences, Heilongjiang University, Harbin 150080, China

Received: 22 July 2023 Revised: 07 October 2023 Accepted: 07 October 2023 Published: 24 October 2023
35K35, 35K65, 35A01, 35D30

In this work, the initial-boundary value problem for the global dynamical properties of solutions to a class of finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied. With the help of the Nehari flow and Levine's concavity method, we establish some sharp-like threshold classifications of the initial data under sub-critical, critical and supercritical initial energy levels, that is, we describe the size of an initial data set. It requires the presumption that the initial data starting from one region of phase space have uniform global dynamical behavior, which means that the solution exists globally and decays via energy estimates that ultimately result in the solution tending to zero in the forward time. For the case in which the initial data corresponds to another region, we prove that the solutions related to these initial data are subject to blow-up phenomena in a finite time. In addition, we estimate the corresponding upper bound of the lifespan of the blow-up solution.
- global existence,
- finite time blow up,
- ground state solution,
- degenerate parabolic equation
Citation: Yuxuan Chen. Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity[J]. Communications in Analysis and Mechanics, 2023, 15(4): 658-694. doi: 10.3934/cam.2023033

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Abstract

In this work, the initial-boundary value problem for the global dynamical properties of solutions to a class of finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied. With the help of the Nehari flow and Levine's concavity method, we establish some sharp-like threshold classifications of the initial data under sub-critical, critical and supercritical initial energy levels, that is, we describe the size of an initial data set. It requires the presumption that the initial data starting from one region of phase space have uniform global dynamical behavior, which means that the solution exists globally and decays via energy estimates that ultimately result in the solution tending to zero in the forward time. For the case in which the initial data corresponds to another region, we prove that the solutions related to these initial data are subject to blow-up phenomena in a finite time. In addition, we estimate the corresponding upper bound of the lifespan of the blow-up solution.

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