Some borderline cases of nonlinear parabolic equations with irregular data

In this paper, we prove existence of solutions for nonlinear parabolic equations whose model is

$$u' - {\rm div} \, (|\nabla u|^{p-2}\nabla u) = f \quad {\rm on} \, \Omega \times (0,T),$$

with homogeneous Cauchy–Dirichlet boundary conditions, where \({1 < p < 2}\). Here f belongs to L ¹ or to L ^m , with m “small.”