| Abstract: |
| We study time-dependent dynamics in driven-damped $\phi^4$ models with spatial and spatiotemporal modulations, focusing on the connection between reduced dynamical descriptions and rigorous PDE-level analysis. Collective-coordinate reductions reveal complex behavior, including separatrix splitting and chaotic kink motion under periodic forcing. However, the existence of time-periodic solutions at the level of the full PDE remains largely open.
We address this by proving the existence of time-periodic solutions for a driven-damped $\phi^4$ equation on a bounded domain with periodic boundary conditions. The analysis is based on a Lyapunov-Schmidt reduction in a time-periodic Hilbert space. A key feature is that damping ensures uniform invertibility of the linearized operator, eliminating the need for non-resonance conditions typical of conservative systems.
Our results provide a rigorous PDE-level counterpart to time-periodic responses observed in reduced models and numerical studies. |
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