Spectrum and stabilization in hyperbolic problems (Q1991904)

The authors studies the mixed problem \[ \begin{gathered} u_{tt}-\Delta u=0,\quad t>0, \quad x\in\Omega, \\ u=0,\quad x\in\partial\Omega, \qquad u(0,x)=f(x),\qquad u_t(0,x)=g(x). \end{gathered} \] Here \(\Omega\subseteq\mathbb{R}^n\), \(n\geqslant 2\), which can be also unbounded. The main aim is to study the relationship between spectral properties of the Laplace operator and the stabilization of the solutions as \(t\to+\infty\). Denote \(\mathcal{E}_{\Omega'}(t):=\|\nabla u\|_{L_2(\Omega')}^2\). The main results are as follows. The first result states that if \[ \lim_{t\to+\infty} \frac{1}{t}\int_{0}^{t}\mathcal{E}_{\Omega'(\tau)}d\tau=0 \tag{1} \] for each \(\Omega'\subset\Omega\) and \(\Omega\) is bounded, then \(u=0\). The second result concerns an unbounded domain \(\Omega\) and it says that, if the point spectrum of the Dirichlet Laplacian in \(\Omega\) is non-empty, then there exist functions \(f, g \in C_0^\infty(\Omega)\) and a bounded domain \(\Omega'\subset\Omega\) such that \[ \liminf_{t\to+\infty} \mathcal{E}_{\Omega'}(t)>0. \] If the point spectrum is empty, then for \(f\in \overset{\circ}{H}^1(\Omega)\), \(g\in L_2(\Omega)\) and bounded domains \(\Omega'\subset\Omega\) we again have (1). If in addition we know that the spectrum of the Dirichlet Laplacian is absolutely continuous, then \[ \lim_{t\to+\infty} \mathcal{E}_{\Omega'}(t)=0. \] The paper also contains a series of theorems providing more detailed information about the decay of \(\mathcal{E}_{\Omega'}(t)\).