The paper is devoted to the study of the homogeneous Dirichlet problem for the doubly nonlinear parabolic equation with nonstandard growth conditions: ut = div ( a(x, t, u)|u|α(x,t)|∇u|p(x,t)−2∇u)+ f (x, t) with given variable exponents α(x, t) and p(x, t). We establish conditions on the data which guarantee the existence of bounded weak solutions in… (More)