Topological and geometric methods for the locally damped supercritical 3D wave equation

We study two distinct but deeply connected regimes for the locally damped wave equation with supercritical nonlinearities in dimension N = 3, highlighting the interplay between higher-order topologies and spatial geometry. In the first part, we consider the equation with a strongly supercritical defocusing nonlinearity u 7 and localized linear damping a ( x ) u t . Due to the supercritical nature of the nonlinearity ( s c = 7/6 > 1), uniqueness of weak solutions in the standard energy space is a challenging open problem. We bypass this topological barrier by elevating the analysis to the higher-order energy space H 2 × H 1 . By exploiting the Gagliardo–Nirenberg interpolation and employing a unified compactness-uniqueness argument on the time-differentiated system, we establish the existence, uniqueness, and uniform exponential stabilization of strong solutions globally in time (for small initial data), circumventing the lack of weak observability. The exponential decay is not a mere consequence of global existence, but the primary mechanism that prevents the finite-time blow-up of the H 2 norm. In the second part, we investigate the focusing regime featuring a localized nonlinear damping a ( x ) u t p − 1 u t a(x) uₜ ᵖ⁻¹uₜ competing against a localized focusing source b ( x )| u | q −1 u . We reveal a novel geometric threshold that supersedes the classical algebraic conditions. We prove that the well-known condition p ≥ q is entirely insufficient to prevent finite-time blow-up if the source operates outside the damping region. By exploiting the finite speed of propagation, we demonstrate a geometric blow-up phenomenon, concluding that the strict inclusion of the source support into the effective damping region is a mandatory requirement for global solvability, regardless of the algebraic dominance of the dissipation.