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This paper is concerned with the semilinear damped wave equation on a measurespace with a self-adjoint operator, instead of the standard Laplace operator.Under a certain decay estimate on the corresponding heat semigroup, weestablish the linear estimates which generalize the so-called Matsumuraestimates, and prove the small data global existence of solutions to the dampedwave equation based on the linear estimates. Our approach is based on a directspectral analysis analogous to the Fourier analysis. The self-adjoint operatorstreated in this paper include some important examples such as the Laplaceoperators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary openset, the Robin Laplacian on an exterior domain, the Schr\"odinger operator, theelliptic operator, the Laplacian on Sierpinski gasket, and the fractionalLaplacian.

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