Existence of weak solutions to an elliptic-parabolic equation with variable order of nonlinearity

We consider an equation with variable nonlinearity of the form $|u|^{p(x)}$, in which the parabolic term can vanish, i.e., in the corresponding domain the parabolic equation becomes “elliptic.” Under a weak monotonicity conditions (nonstrict inequality) we prove the existence of a solution to the first mixed problem in a cylinder with a bounded base.