In this paper, we study a Cauchy problem for the nonlinear damped wave equations for a general positive operator with discrete spectrum. We derive the exponential in time decay of solutions to the linear problem with decay rate depending on the interplay between the bottom of the operator's spectrum and the mass term. Consequently, we prove global in time well-posedness results for semilinear and for more general nonlinear equations with small data. Examples are given for nonlinear damped wave equations for the harmonic oscillator, for the twisted Laplacian (Landau Hamiltonian), and for the Laplacians on compact manifolds.

中文翻译：

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关于正算符的非线性阻尼波方程。一，离散频谱

在本文中，我们研究了具有离散谱的一般正算子的非线性阻尼波方程的柯西问题。我们推导了线性方程解的时间衰减的指数，其衰减率取决于操作员频谱底部与质量项之间的相互作用。因此，我们证明了半线性方程组和时间更短的非线性方程组（具有少量数据）在时间上的适定性结果。给出了用于谐波振荡器，扭曲的拉普拉斯算子（Landau Hamiltonian）和紧凑流形上的拉普拉斯算子的非线性阻尼波方程的示例。