Estimates of decay rate for solution to parabolic equation with non-power nonlinearities

We study the Dirichlet mixed problem for a class parabolic equation with double non-power nonlinearities in cylindrical domain $D=(t>0)\times\Omega$. By the Galerkin approximations method suggested by Mukminov F. Kh. for a parabolic equation with double nonlinearities we prove the existence of strong solutions in Sobolev–Orlicz space. The maximum principle as well as upper and lower estimates characterizing powerlike decay of solution as $t\to\infty$ in bounded and unbounded domains $\Omega\subset R_n$ are established.