Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.20614

Time-Periodic Solutions of a Driven--Damped $ϕ^4$ Equation on Bounded Domains

Vassilis M Rothos
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove the existence of time-periodic solutions for a driven--damped $ϕ^4$ wave equation posed on a bounded spatial domain with periodic boundary conditions. The result is obtained at the level of the full partial differential equation. The analysis is based on a Lyapunov--Schmidt decomposition in a time-periodic function space. A key structural feature is the presence of linear damping, which ensures uniform invertibility of the associated linear operator on the complement of the kernel and removes the need for nonresonance conditions. After introducing a truncation to control the cubic nonlinearity, we solve the complement equation via a contraction mapping argument and reduce the problem to a scalar kernel equation. The latter is treated using a mean/zero-mean decomposition and a fixed-point argument. A period-averaged energy estimate then allows one to remove the truncation and obtain a time-periodic weak solution under explicit smallness conditions on the forcing parameters

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vassilis M Rothos. 2026-05-23. Time-Periodic Solutions of a Driven--Damped $ϕ^4$ Equation on Bounded Domains. https://arxiv.org/abs/2606.20614

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A linear test approach to global controllability of third- and fifth-order nonlinear dispersive equations

We investigate third- and fifth-order nonlinear dispersive equations of KdV type on the torus and establishes approximate controllability by a fixed four-dimensional control; rather than relying solely on the saturation machinery, the analysis exploits the finite-dimensional controllability of the inviscid Burgers equation linearized around a carefully constructed return trajectory, with the trajectory itself obtained from an observable family. This ``linear test" strategy, yields more information about the structure of the control than the standard approach. In particular, the constructed control is shown to depend continuously on the initial and target states, a property that is by no means automatic in nonlinear control problems, and to decompose as a bounded linear operator applied to the data plus a fixed remainder, with the operator part interestingly independent of the order of dispersion.

math.AP↗
arXiv2026

A conditional Lagrangian clock barrier at the $C^{1,\frac{1}{3}}$ threshold for axisymmetric Euler without swirl

We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,α}\cap L^2$, where $α\in\left[\frac{1}{3},1\right)$. In a major breakthrough, Shkoller introduced a clock-and-driver framework that he used in order to prove finite-time type I blow-up below the $C^{1,\frac{1}{3}}$ threshold in this setting. Motivated by this, we define Lagrangian classes of coherent conditional solutions for which the same mechanism yields a supercritical-critical barrier to blow-up when $α\geq\frac{1}{3}$. When $α>\frac{1}{3}$, the aforementioned barrier is genuinely depleted, whereas at the critical endpoint $α=\frac{1}{3}$, we obtain an exponential bound preventing blow-up. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under transverse cusp-tail, longitudinal, off-clock, Dini, and suitable geometric coherence hypotheses, this singular value cannot collapse in finite time. In particular, we also show that the class of such coherent solutions includes the smooth ones locally in time. In the on-axis case, the criterion reduces to the scalar clock inequality $\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t)^{3α}$, which rules out Shkoller-type clock collapse for $α\geq\frac{1}{3}$. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller's subcritical blow-up mechanism in the case $α>\frac{1}{3}$.

math.AP↗