We study the asymptotic behavior of the solution of semi-linear PDEs. Neither periodicity nor ergodicity assumptions are assumed. The coefficients admit only a limit in a Cesaro sense. In such a case, the limit coefficients may have discontinuity. We use probabilistic approach based on weak convergence techniques for the associated backward stochastic differential equation in the S-topology. We establish weak continuity for the flow of the limit diffusion process and related the PDE limit to the backward stochastic differential equation via the representation of Lp-viscosity solution.