Strong Galerkin Approximation, Malliavin Regularity, and Blow-Up for a Mixed Local--Nonlocal Stochastic Wave Equation

Abstract

We investigate the dynamical behavior of a class of semilinear stochastic wave equations on a bounded smooth domain $\Ocal\subset\R^d$ driven by additive trace-class noise, where the elastic response is governed by a \emph{mixed local--nonlocal} operator $\Acal=-\theta\Delta+\beta(-\Delta)^s$ with $s\in(0,1)$. A fundamental challenge in this setting is that the local and nonlocal operators do not commute on bounded domains: the natural Dirichlet basis fails to diagonalize the restricted fractional Laplacian. Consequently, we first establish the \emph{strong} convergence of the resulting non-diagonal, dense Galerkin approximation scheme. Leveraging these uniform energy bounds, we rigorously derive the associated It\^{o} energy identity. In the defocusing regime ($\varepsilon = +1$), this strong approximation yields global well-posedness on the energy-subcritical range, providing a unique probabilistically strong solution in the energy space $V \times H$. Within this variational framework, we conduct an analysis of the Malliavin regularity, showing $(u(t), v(t)) \in \mathbb{D}^{1,2}(V) \times \mathbb{D}^{1,2}(H)$, and leverage fractional Sobolev embeddings to prove that the one-dimensional probability law of $u(t, x_0)$ is absolutely continuous via the Bouleau--Hirsch criterion. In stark contrast, for the focusing regime ($\varepsilon = -1$), we establish local well-posedness and prove a rigorous dichotomy: under a negativity condition on the initial energy, either pathwise explosion occurs with positive probability in finite time, or the energy norm possesses an infinite second moment before an explicit critical time $T^*$. Finally, we observe how the dense Galerkin interaction matrices pose unique structural challenges for spatial statistical inference.