Regular Solutions for 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 high-order sources have been a major challenge in the investigation of finite-energy (H1 × L2) solutions to wave PDEs for many years. The well-posedness question has been answered in part, but even the local existence, for instance, in 3 dimensions requires the relation p ≤ 6m/(m + 1) between the exponents p of the source and m of the viscous damping. We prove that smoother initial data (H2 × H1) yields regular solutions that do not necessitate a correlation of the source and the damping. Local existence of such solutions is shown for any source exponent p ≥ 1 and any monotone damping including: exponential, logarithmic, or none at all in dimensions 3 and 4 (and with some restrictions on p in dimensions n ≥ 5). This result extends the known theory which in the context of supercritical sources predominantly focuses on damping of polynomial growth and guarantees local smooth solutions without correlating the damping and the source only if p < 5 in 3D (or p < n+2 n−2 if n = 3, 4). Furthermore, if we assert the classical condition that the damping grows at least as fast the source, then these regular solutions are global.