Existence, uniqueness, and stability of stochastic wave equation with cubic nonlinearities in two dimensions

The main focus of this article is the qualitative and quantitative behavior of stochastic wave equations with cubic nonlinearities in two dimensions. We prove that the strong Fourier solution of these semi-linear wave equation exists and is unique on an appropriate Hilbert space. Also, we study the stability of solutions and give conclusions in three cases: stability in probability, estimates of LpLp-growth, and almost sure exponential stability. The main tool is the study of related Lyapunov-type functionals which admits to control the total energy of randomly vibrating membranes. The analysis is carried out by a natural N-dimensional truncation in isometric Hilbert spaces and uniform estimation of moments with respect to N.