Decay of solutions of the wave equation and spectral properties of the Laplace operator in expanding domains
Let \(\Omega\subset \mathbb{R}^n\), \(n\geq 2\), be an unbounded domain with boundary \(\Gamma\in C^2\). Consider the problem \[ u_{tt}- \Delta u= 0,\quad t>0,\quad x\in\Omega,\quad u(0, x)= f(x),\quad u_t(0, x)= g(x),\quad u|_\Gamma= 0,\tag{1} \] where \(f(x)\in \mathring W^1_2(\Omega)\) and \(g(x)\in L_2(\Omega)\) are real-valued initial functions. We say that the energy \[ E_\Omega(t)= \int_\Omega (u^2_t+|\nabla u|^2)dx= E_\Omega(0),\quad t>0 \] scatters to infinity if for any bounded domain \(\Omega'\subset \Omega\) we have \(\lim_{t\to\infty} E_{\Omega'}(t)= 0\), where \[ E_{\Omega'}(t)= \int_{\Omega'} (u^2_t+|\nabla u|^2)dx. \] We are interested in the relation between the scattering of energy to infinity for the solution of problem (1) and the spectral properties of the associated elliptic problem.
scientific article; zbMATH DE number 1228266
scientific article