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 $\Acal=-θΔ+β(-Δ)^s$ 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 Itô energy identity. In the defocusing regime ($\epsilon =+1$), 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 ($\epsilon =-1$), 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 $T^{\ast }$. Finally, we observe how the dense Galerkin interaction matrices pose unique structural challenges for spatial statistical inference.

27 Pages