Differential and Integral Equations

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

Michael Ruzhansky and Niyaz Tokmagambetov

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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.

Article information

Source
Differential Integral Equations, Volume 32, Number 7/8 (2019), 455-478.

Dates
First available in Project Euclid: 2 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.die/1556762425

Mathematical Reviews number (MathSciNet)
MR3945764

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35L05: Wave equation 35L70: Nonlinear second-order hyperbolic equations 42A85: Convolution, factorization 35P10: Completeness of eigenfunctions, eigenfunction expansions 44A35: Convolution

Citation

Ruzhansky, Michael; Tokmagambetov, Niyaz. On nonlinear damped wave equations for positive operators. I. Discrete spectrum. Differential Integral Equations 32 (2019), no. 7/8, 455--478. https://projecteuclid.org/euclid.die/1556762425


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