GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR SEMILINEAR DAMPED WAVE EQUATIONS ON MEASURE SPACES

The purpose of this paper is to prove the small data global existence of solutions to the semilinear damped wave equation partial derivative(2)(t)u + Au + partial derivative(t)u = |u|(p-1)u on a measure space X with a self-adjoint operator A on L-2(X). Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura 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 Schrodinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian.