Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension

Abstract

The aim of this paper is to prove that, for specific initial data $(u_0,u_1)$ and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen-Cahn equation on the interval $[a,b]$ shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the "energy approach" proposed by Bronsard and Kohn [8], if $\epsilon \ll 1$ is the diffusion coefficient, we show that in a time scale of order ${\epsilon }^{-k}$ nothing happens and the solution maintains the same number of transitions of its initial datum $u_0$. The novelty consists mainly in the role of the initial velocity $u_1$, which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen-Cahn equation with relaxation.

References in corpus (2)

• Analytical and numerical investigation of traveling waves for the Allen-Cahn model with relaxation
• Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems

Cited by in corpus (9)

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• Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms
• Slow motion for one-dimensional nonlinear damped hyperbolic Allen-Cahn systems
• Metastability and layer dynamics for the hyperbolic relaxation of the Cahn-Hilliard equation
• Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator
• Spectral stability of traveling fronts for nonlinear hyperbolic equations of bistable type
• Motion of interfaces for a damped hyperbolic Allen-Cahn equation
• Metastable dynamics for a hyperbolic variant of the mass conserving Allen-Cahn equation in one space dimension