On nonlinear damped wave equations for positive operators. I: Discrete spectrum. (Q2280476)

The work is devoted to the analysis of the Cauchy problem to the equation \(u_{tt}(t)+\mathcal{L}u(t)+bu_t(t)+mu(t)=F(u,u_t,\mathcal{L}^{1/2}u)\), where \(b>0\) and \(\mathcal{L}\) is a positive self-adjoint operator in a Hilbert space with a discrete spectrum. Under some assumptions on \(b,m,\mathcal{L}\) and the nonlinearity \(F\), the existence of the unique solution and its exponential decay for small initial data are proved. The authors present the harmonic oscillator, twisted Laplacian (Landau Hamiltonian) and the Laplacians on compact manifolds as examples.