Lebesgue Space Estimates for Solutions to Hyperbolic Equations

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9706840 Beals The investigation involves the study of certain properties of solutions to strictly hyperbolic partial differential equations ("waves"). New qize and smoothness estimates in the linear case will be derived, in order to analyze the properties of solutions to nonlinear problems. Estimates of time decay for appropriate measures of the size of a solution will be obtained. Results for the classical wave equation on a spatial domain which is exterior to a convex obstacle will be extended to handle more general natural conditions at the boundary of the obstacle. The key estimates correspond to finding the best possible time decay subject to the weakest size and smoothness restrictions on the data. More general obstacles, for which certain types of decay estimates are known but for which explicit representations of approximations to solutions are absent, will also be treated. Analogous full-space problems (no obstacles) will be analyzed. The primary questions in these cases involve higher order equations or systems, and the effects of interactions with changes in the properties of the medium in which the waves are propagating. Nonsmooth perturbations in the medium are one source of problems, including questions about the minimal regularity needed for the perturbing potential. Propagation in inhomogeneous media (for example, nonconstant wave speeds) where bounds involving explicit time estimates are relatively sparse, will also be considered. (The classical bounds involve those which measure size in terms of the classical energy, which will not typically exhibit decay in time.) The optimal time estimates under the best possible conditions on the size of the data measured in terms of other natural spaces will be considered, in model problems and in general. Waves have been studied classically in terms of a size quantity like energy, which is typically conserved or behaves in a fashion which is easy to understand. In order to go beyond the simpl est mathematical approximations to understand the small- and large-scale properties of true physical waves, other means of measuring the wave size must be used. Only the simplest model problems have been fully understood in this context in the past - those for which the large-scale behavior is merely a simple magnification of the small-scale behavior (such as unperturbed waves in free space). In true physical problems, waves encounter obstacles and are reflected or absorbed; they travel in media which are not uniform; they interact with themselves and with the media in which they propagate. The size behavior of such waves, and in particular the decay of their amplitudes in time, will be analyzed, in order to understand what information can be deduced about the wave source from measurements of such size quantities made far away (or long after the wave is generated).