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On nonlinear damped wave equations for positive operators. I: Discrete spectrum. (English) Zbl 1463.35353

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.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L05 Wave equation
42A85 Convolution, factorization for one variable harmonic analysis
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
44A35 Convolution as an integral transform
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