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We establish the existence and uniqueness of weak solutions to the parabolic system with nonstandard growth condition and cross diffusion, $$\displaylines{ \partial_tu-\text{div}a(x,t,\nabla u)) =\text{div}|F|^{p(x,t)-2}F),\cr \partial_tv-\text{div}a(x,t,\nabla v))=\delta\Delta u, }$$ where $\delta\ge0$ and $\partial_tu,~\partial_tv$ denote the partial derivative of u and v with respect to the time variable t, while $\nabla u$ and $\nabla v$ denote the one with respect to the spatial variable x. Moreover, the vector field $a(x,t,\cdot)$ satisfies certain nonstandard p(x,t) growth, monotonicity and coercivity conditions.