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 $${\vartheta }_{tt}+\gamma {\vartheta }_t-\Delta \vartheta +|\vartheta {|}^{p-2}\vartheta =0$$ on a smooth bounded domain $\Omega \subset {\mathbb{R}}^d$, where the evolution is projected onto the tangent space of the Hilbert manifold $$\mathcal{M}=\left\{\vartheta \in L^2(\Omega ):\Vert \vartheta {\Vert }_{L^2(\Omega )}=1\right\},$$ which is the unit sphere in $L^2(\Omega )$. We assume that $$p\in [2,\infty )\text{ for }d=1,2,\text{ while }2\le p\le \frac{2(d-1)}{d-2}\text{ for }d\ge 3.$$ 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 $\mathcal{M}$ 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 $H_0^1(\Omega )$ to the unique positive ground-state solution.