Existence, uniqueness, and stability of Fourier series solutions of stochastic wave equations with cubic nonlinearities in 3D

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) utt=σ2△u+B(u,ut)+G(u,ut)∂W∂t with cubic nonlinearities B(u,ut)=a1u−a2||u||2u−κut and homogeneous boundary conditions (HBCs) on general 3D cubes D=[0,lx]×[0,ly]×[0,lz] is studied. The driving Q-regular space–time noise W with linear-growth bounded, state-dependent diffusion intensities G(u,ut) 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 a1, length parameters lx,ly,lz, diffusion constants br of G and eigenvalues αn,m,li,j,k of related covariance operator Q. Several examples illustrate the feasibility of our approach and explain our major results.