  We study the asymptotic behavior of solution of semi-linear PDEs. Neither
periodicity nor ergodicity will be assumed. In return, we assume that the
coefficients admit a limit in \`{C}esaro sense. In such a case, the averaged
coefficients could be discontinuous. We use probabilistic approach based on
weak convergence for the associated backward stochastic differential equation
in the S-topology to derive the averaged PDE. However, since the averaged
coefficients are discontinuous, the classical viscosity solution is not defined
for the averaged PDE. We then use the notion of "$L^p-$viscosity solution"
introduced in \cite{CCKS}. We use BSDEs techniques to establish the existence
of $L^p-$viscosity solution for the averaged PDE. 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
$L^p$-viscosity solution.
