REGULAR SOLUTIONS OF WAVE EQUATIONS WITH SUPER-CRITICAL SOURCES AND EXPONENTIAL-TO-LOGARITHMIC DAMPING

We study regular solutions to wave equations with super-critical source terms, e.g., of exponent p > 5 in 3D. Such sources have been a major challenge in the investigation of finite-energy (H-1 x L-2) solutions to wave PDEs for many years. The wellposedness has been settled in part, but even the local existence, for instance, in 3 dimensions requires the relation p <= 6 m/(m + 1) between the exponents p of the source and m of the viscous damping. We prove that smooth initial data (H-2 x H-1) yields regular solutions that do not depend on the above correlation. Local existence is demonstrated for any source exponent p >= 1 and any monotone damping including feedbacks growing exponentially or logarithmically at infinity, or with no damping at all. The result holds in dimensions 3 and 4, and with some restrictions on p in dimensions n >= 5. Furthermore, if we assert the classical condition that the damping grows as fast as the source, then these regular solutions are global.