WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION

doi:10.1007/s10473-025-0411-4

Abstract

Abstract: In this paper, we consider the initial boundary value problem for the 2-D hyperbolic viscous Cahn-Hilliard equation. Firstly, we prove the existence and uniqueness of the local solution by the Galerkin method and contraction mapping principle. Then, using the potential well theory, we study the global well-posedness of the solution with initial data at different levels of initial energy, i.e., subcritical initial energy, critical initial energy and arbitrary positive initial energy. For subcritical initial energy, we prove the global existence, asymptotic behavior and finite time blowup of the solution. Moreover, we extend these results to the critical initial energy using the scaling technique. For arbitrary positive initial energy, including the sup-critical initial energy, we obtain the sufficient conditions for finite time blow-up of the solution. As a further study for estimating the blowup time, we give a unified expression of the lower bound of blowup time for all three initial energy levels and estimate the upper bound of blowup time for subcritical and critical initial energy.

Key words: 2-D hyperbolic viscous Cahn-Hilliard equation, global existence, finite time blow up, exponential decay

Siyan GUO, Jiangbo HAN, Runzhang XU. WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION[J].Acta mathematica scientia,Series B, 2025, 45(4): 1438-1470.

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