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 disc
ontinuous. 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.
We study multidimensional backward stochastic differential equations (BSDEs) which cover the logarithmic nonlinearity u log u. More precisely, we establish the existence and uniqueness as well as the stability of p-integrable solutions (p > 1) to mul
tidimensional BSDEs with a p-integrable terminal condition and a super-linear growth generator in the both variables y and z. This is done with a generator f(y, z) which can be neither locally monotone in the variable y nor locally Lipschitz in the variable z. Moreover, it is not uniformly continuous. As application, we establish the existence and uniqueness of Sobolev solutions to possibly degenerate systems of semilinear parabolic PDEs with super-linear growth generator and an p-integrable terminal data. Our result cover, for instance, certain (systems of) PDEs arising in physics.
