On nonlinear damped wave equations for positive operators. I. Discrete spectrum

TL;DR

This paper investigates the decay behavior and well-posedness of nonlinear damped wave equations involving positive operators with discrete spectra, providing new insights into their long-term dynamics and solutions.

Contribution

It introduces decay estimates and global well-posedness results for nonlinear damped wave equations with positive operators, including specific examples like the harmonic oscillator and Landau Hamiltonian.

Findings

Solutions exhibit exponential decay depending on spectral properties.

Global existence results are established for small initial data.

Examples include harmonic oscillator and Laplacians on manifolds.

Abstract

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.