Approximation of the initial value for damped nonlinear hyperbolic equations with random Gaussian white noise on the measurements

The main goal of this work is to study a regularization method to reconstruct the solution of the backward non-linear hyperbolic equation $u_{tt}+\alpha {\mathrm{\Delta }}^2{u}_t+\beta {\mathrm{\Delta }}^{2}u=\mathcal{F}(x,t,u)$ come with the input data are blurred by random Gaussian white noise. We first prove that the considered problem is ill-posed (in the sense of Hadamard), i.e., the solution does not depend continuously on the data. Then we propose the Fourier truncation method for stabilizing the ill-posed problem. Base on some priori assumptions for the true solution we derive the error and a convergence rate between a mild solution and its regularized solutions. Also, a numerical example is provided to confirm the efficiency of theoretical results.