[Paper Review] Global behaviour of bistable solutions for hyperbolic gradient systems in one unbounded spatial dimension

Emmanuel Risler|arXiv (Cornell University)|Mar 3, 2017

Stability and Controllability of Differential Equations6 references3 citations

TL;DR

This paper establishes the global asymptotic behavior of bistable solutions for damped hyperbolic gradient systems in one unbounded spatial dimension, proving that solutions converge to a left-traveling and right-traveling stack of bistable fronts, with a central region exhibiting a slowly evolving pattern of stationary solutions homoclinic or heteroclinic to stable equilibria. The analysis relies on purely variational methods due to the absence of a maximum principle, extending prior results on parabolic and damped wave systems to the hyperbolic case.

ABSTRACT

This paper is concerned with damped hyperbolic gradient systems of the form \\[ \\alpha u_{tt} + u_t = -\ abla V(u) + u_{xx}\\,, \\] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, $\\alpha$ is a positive quantity, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of every bistable solution (that is, every solution close at both ends of space to stable homogeneous equilibria) is described. Every such solution approaches, far to the left in space a stacked family of bistable fronts travelling to the left, far to the right in space a stacked family of bistable fronts travelling to the right, and in between a pattern of profiles of stationary solutions homoclinic or heteroclinic to stable homogeneous equilibria, going slowly away from one another. In the absence of maximum principle, the arguments are purely variational. This extends previous results obtained in companion papers for damped wave equations or parabolic gradient systems, in the spirit of the program initiated in the late seventies by Fife and McLeod on the global asymptotic behaviour of bistable solutions for parabolic equations.

Motivation & Objective

• To describe the long-time asymptotic behavior of bistable solutions in damped hyperbolic gradient systems on the whole real line.
• To extend previous results on parabolic and damped wave equations to the hyperbolic case, where the maximum principle does not hold.
• To characterize the spatial structure of solutions as a combination of left-moving and right-moving traveling fronts and a central pattern of stationary solutions.
• To establish convergence to a propagating terrace of traveling fronts and a standing terrace of stationary solutions under generic assumptions on the potential.

Proposed method

• Uses a variational framework to analyze the energy and dissipation of solutions in uniformly local Sobolev spaces.
• Introduces a 'firewall function' in both laboratory and traveling frames to control invasion speeds and energy decay.
• Applies a 'relaxation scheme' to track energy decrease and pollution effects in localized energy estimates.
• Employs a 'smooth plus small' decomposition to analyze convergence to stationary profiles in the central region.
• Relies on coercivity and generic hypotheses on the potential, including escape distance and breakup of translation invariance.
• Establishes convergence via energy-based estimates and time-averaged dissipation control, avoiding maximum principle arguments.

Experimental results

Research questions

• RQ1 How do bistable solutions of damped hyperbolic gradient systems behave globally in one unbounded spatial dimension?
• RQ2 What is the asymptotic structure of such solutions—specifically, do they decompose into traveling fronts and stationary patterns?
• RQ3 How can the dynamics be controlled in the absence of a maximum principle, given the hyperbolic nature of the system?
• RQ4 What conditions ensure that the solution approaches a propagating terrace of traveling fronts and a standing terrace of stationary solutions?
• RQ5 How does the energy of the solution evolve, and what is the role of the residual asymptotic energy in the long-time limit?

Key findings

• Every bistable solution approaches, far to the left, a stacked family of bistable fronts traveling to the left, and far to the right, a stacked family of fronts traveling to the right.
• In the central spatial region, the solution exhibits a slowly evolving pattern of stationary solutions that are homoclinic or heteroclinic to stable homogeneous equilibria.
• The time derivative of the solution converges to zero in the central region, indicating a slow evolution of the profile.
• The residual asymptotic energy of the solution equals the energy of the standing terrace of stationary solutions, confirming energy conservation in the limit.
• The invasion speed is bounded from above and converges to a limit, with subsonic bounds established via firewall function analysis.
• Convergence to the standing terrace is achieved in the normalized Hamiltonian level set zero, with the solution profile approaching the set of bistable stationary solutions.