Existence, uniqueness, and stability of stochastic wave equation with cubic nonlinearities in two dimensions
Assume that \(l_x\) and \(l_y\) are two positive real numbers and \(D=[0,l_x]\times [0,l_y]\). The aim of this paper is to study the nonlinear wave equation \[ u_{tt}=\sigma^2( u_{xx}+ u_{yy}) + A(u,v) + B(u,v) \frac{dW}{dt} \] in which \(A(u,v)= (a_1 - a_2\|u\|^2_{L^2(D)})u -\kappa v \) and \(B(u,v)= b_0 +b_1\|u\|_{L^2(D)} + b_2\|v\|_{L^2(D)}\). The authors assume a two-dimensional Fourier series expansion for the solution and prove that the coefficients satisfy an ordinary stochastic differential equation. Then, they define a finite-dimensional truncated Fourier series and Lyapunov functional for this equation to obtain an existence and uniqueness result for a continuous strong solution. Since the estimates for the Lyapunov functionals of the approximating systems are uniform with respect to the dimension \(N\), the authors can obtain estimates for the total expected energy of the above equation. Moreover, they prove the moment and sample path stability of the solutions by the method of Lyapunov functionals and give a bound for the Lyapunov exponent.
scientific article; zbMATH DE number 6390525
scientific article