Existence of weak solutions for a class of non-divergent parabolic equations with variable exponent

A doubly degenerate parabolic equation in non-divergent form with variable growth is investigated in this paper. In suitable spaces, we prove the existence of weak solutions of the equation for cases $1\le m<2$ and $m\ge 2$ in different ways. And we establish the non-expansion of support of the solution for the problem.