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Pullback Attractors for Nonclassical Diffusion Equations in Noncylindrical Domains

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International Journal of Mathematics and Mathematical Sciences
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Abstract

The existence and uniqueness of a variational solution are proved for the following nonautonomous nonclassical diffusion equation 𝑒 𝑑 βˆ’ πœ€ Ξ” 𝑒 𝑑 βˆ’ Ξ” 𝑒 + 𝑓 ( 𝑒 ) = 𝑔 ( π‘₯ , 𝑑 ) , πœ€ ∈ ( 0 , 1 ] , in a noncylindrical domain with homogeneous Dirichlet boundary conditions, under the assumption that the spatial domains are bounded and increase with time. Moreover, the nonautonomous dynamical system generated by this class of solutions is shown to have a pullback attractor  π’œ πœ€ , which is upper semicontinuous at πœ€ = 0 .

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