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)=u₀(x), where v>0 is a fixed constant, t≥0, x∈Ω, Ω is a bounded domain in Rⁿ. 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.