Asymptotic behaviour of solutions to some pseudoparabolic equations

The aim of this paper is to investigate the behaviour as t → ∞ of solutions to the Cauchy problem u t − △ u t − v △ u − ( b , ∇ u ) = ∇ ⋅ F ( u ) , u ( x , 0 ) = u 0 ( x ) , where v > 0 is a fixed constant, t ≥ 0 , x ∈ Ω , Ω is a bounded domain in R n . We will first establish an a priori estimate. Then, we establish the global existence, uniqueness and continuous dependence of the weak solution for the Sobolev–Galpern type equation with the Dirichlet boundary.