Abstract: The paper is devoted to a certain class of doubly nonlinear higher-order anisotropic parabolic equations Using Galerkin approximations it is proved that the first mixed problem with homogeneous Dirichlet boundary condition has a strong solution in the cylinder $D=(0 \infty)\times\Omega$  where $\Omega\subset\mathbb R^n$  $n\geq 3$  is an unbounded domain