Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

TL;DR

This paper extends the analysis of semilinear damped wave equations to measure spaces with self-adjoint operators, establishing global existence and asymptotic behavior under decay estimates, using spectral analysis methods.

Contribution

It generalizes Matsumura estimates to measure spaces with various self-adjoint operators, enabling broader application to different geometric and fractal domains.

Findings

Established linear estimates generalizing Matsumura estimates

Proved small data global existence of solutions

Applied spectral analysis to diverse operators

Abstract

This paper is concerned with the semilinear damped wave equation on a measure space with a self-adjoint operator, instead of the standard Laplace operator. Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates, and prove the small data global existence of solutions to the damped wave equation based on the linear estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schr\"odinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian.