Upper semicontinuity of pullback D$\mathcal {D}$‐attractors for nonlinear parabolic equation with nonstandard growth condition

This paper is devoted to the well‐posed problem and the existence of pullback D$\mathcal {D}$‐attractors for a class of nonlinear parabolic equation with nonstandard growth condition. First, by making use of Galerkin's method and monotone operator method, the existence of solutions is proved in Orlicz–Sobolev space with variable exponents depending on time and space, then the uniqueness and continuity of solutions are also obtained. Finally, by verifying the pullback D$\mathcal {D}$‐asymptotic compactness, the existence of pullback D$\mathcal {D}$‐attractors is proved. In particular, the upper semicontinuity of pullback D$\mathcal {D}$‐attractors of the corresponding equation with respect to the disturbance parameter λ is also proved.