p-integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities

Abstract: 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 multidimensional 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.