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In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation∂ t  u − div a  ( x , t , ∇ u )  +   λ ( | u |   p ( x , t ) − 2    u ) = 0  in    Ω T  ,whereλ ≥ 0and∂ t  udenote the partial derivative of u with respect to the time variable t, while∇ udenotes the one with respect to the space variable x. Moreover, the vector-fielda ( x , t , · )satisfies certain nonstandardp ( x , t )    -growth and monotonicity conditions. In this manuscript, we establish the existence of a unique weak solution to the corresponding Dirichlet problem. Furthermore, we prove the stability of this solution, i.e., we show that two weak solutions with different initial values are controlled by these initial values

The Stability of Parabolic Problems with Nonstandard p(x, t)-Growth