Existence, uniqueness, and stability of Fourier series solutions of stochastic wave equations with cubic nonlinearities in 3D. (February 2023) - Schurz, Henri; Alharbi, Qasim S.
Abstract: Existence, uniqueness, and stability of approximate Fourier series solutions u to damped and undamped, semi-linear stochastic wave equations of the form (in Itô integral sense) u t t = σ 2 △ u + B ( u, u t ) + G ( u, u t ) ∂ W ∂ t with cubic nonlinearities B ( u, u t ) = a 1 u − a 2 | | u | | 2 u − κ u t and homogeneous boundary conditions (HBCs) on general 3D cubes D = [ 0, l x ] × [ 0, l y ] × [ 0, l z ] is studied. The driving Q -regular space–time noise W with linear-growth bounded, state-dependent diffusion intensities G ( u, u t ) is supposed to be general in space ( x, y, z ) ∈ D, but white in time t ≥ 0 . The analysis is carried out on an appropriate, separable Hilbert space (i.e. the space of Fourier coefficients) of all solutions which are driven by the eigenfunctions of Laplace operator △ subject to HBCs on the 3D domain D . The dynamics of related expected total energy functional e ( t ) plays a central role in our discussion, depending on diverse parameters such as diffusivity constant σ, damping κ ≥ 0, transport coefficient a 1, length parameters l x, l y, l z, diffusion constants b r of G and eigenvalues α n, m, l i, j, k of related covariance operator Q . Several examples illustrate the feasibility of our approach and explain our major results.