The purpose of this article is to provethe existence anduniquenessof weak solutionsfornonlinearparabolic problem whose model is \begin{align*} \begin{cases} \frac{ tial v}{ tial t}-peratorname{div}eft[|abla v-heta(v)|^{q(x)-2}(abla v-heta(v))\right]+\beta(v)=f & ext { in }\quad Q_{T}:=(0, T) imes mega , v=0 & ext { on }\quad \Sigma_{T}:=(0, T) imes tial mega, v(\cdot, 0)=v_{0} & ext { in }\quad mega. \end{cases} \end{align*} Wetransform the parabolic problem into the elliptic problem by using time discretization technique by Euler forward scheme and Rothe method combined with the theory of variable exponent Sobolev spaces.