Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint
Abstract
We prove the global existence and uniqueness of strong solutions to a constrained version of the damped nonlinear wave equation ϑtt+γϑt−Δϑ+|ϑ|p−2ϑ=0 on a smooth bounded domain varOmega⊂ℝd, where the evolution is projected onto the tangent space of the Hilbert manifold ℳ = { ϑ∈ L2(varOmega):|ϑ|L2(varOmega)=1 }, which is the unit sphere in L2(varOmega). We assume that p∈[2,∞) for d=1,2, while 2≤ p≤ (2(d−1))/(d−2) for d≥3. By employing the Faedo--Galerkin approximation method, together with suitable a priori estimates and compactness arguments, we establish the global well-posedness of the problem. In particular, we show that the Hilbert manifold ℳ is invariant under the flow, and hence the L2-constraint is preserved throughout the evolution. Using the \emph{Lusternik--Schnirelmann theory}, we show that the corresponding stationary problem possesses at least countably many stationary solutions. We further investigate the long-time behaviour of solutions and prove that, along a subsequence, every solution of the constrained problem converges to a stationary solution by invoking \emph{Webb's theorem} and \emph{Barbalat's lemma}. When the initial data are sufficiently close to the first eigenfunction of the associated stationary problem, we show that the unique strong solution converges in H01(varOmega) to the unique positive ground-state solution.
Related
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BibTeX
@misc{tiwari2026global,
title = {{Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint}},
author = {Harsh Tiwari and Manil T. Mohan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02842},
eprint = {2609.02842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02842},
note = {CC0},
}
RIS
TY - UNPB
TI - Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint
AU - Harsh Tiwari
AU - Manil T. Mohan
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02842
UR - https://arxiv.org/abs/2609.02842
AB - We prove the global existence and uniqueness of strong solutions to a constrained version of the damped nonlinear wave equation ϑ_tt+γϑ_t−Δϑ+|ϑ|^p−2ϑ=0 on a smooth bounded domain varOmega⊂ℝ^d, where the evolution is projected onto the tangent space of the Hilbert manifold ℳ = { ϑ∈ L^2(varOmega):|ϑ|_L^2(varOmega)=1 }, which is the unit sphere in L^2(varOmega). We assume that p∈[2,∞) for d=1,2, while 2≤ p≤ (2(d−1))/(d−2) for d≥3. By employing the Faedo--Galerkin approximation method, together with suitable a priori estimates and compactness arguments, we establish the global well-posedness of the problem. In particular, we show that the Hilbert manifold ℳ is invariant under the flow, and hence the L^2-constraint is preserved throughout the evolution. Using the \emph{Lusternik--Schnirelmann theory}, we show that the corresponding stationary problem possesses at least countably many stationary solutions. We further investigate the long-time behaviour of solutions and prove that, along a subsequence, every solution of the constrained problem converges to a stationary solution by invoking \emph{Webb's theorem} and \emph{Barbalat's lemma}. When the initial data are sufficiently close to the first eigenfunction of the associated stationary problem, we show that the unique strong solution converges in H_0^1(varOmega) to the unique positive ground-state solution.
ER -