On some eigenvalue problems related to the wave equation with indefinite damping
The author considers the damped wave equation of the form \[ u_{tt}+ \varepsilon a(x)u_t=\Delta u+b(x)u \qquad \text{for} \quad x\in\Omega, \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), and solutions are supposed to satisfy homogeneous Dirichlet boundary conditions. The novelty of the paper lies in the fact that the function \(a\) is allowed to change sign on \(\Omega\), specifically, the sets \(\Omega_+\), \(\Omega_-\) on which \(a\) is positive, negative respectively are of positive measure. The rather unexpected main result then states that the zero solution is unstable provided \(\varepsilon\geq \varepsilon_0\) is large enough. The proof is based on careful spectral analysis of the problem.
scientific article; zbMATH DE number 881712