Asymptotic behaviour of solutions to some pseudoparabolic equations

The aim of this paper is to investigate the behaviour as t→∞t→∞ of solutions to the Cauchy problem ut−△ut−v△u−(b,∇u)=∇⋅F(u)ut−△ut−v△u−(b,∇u)=∇⋅F(u),u(x,0)=u0(x)u(x,0)=u0(x), where v>0v>0 is a fixed constant, t≥0t≥0, x∈Ωx∈Ω, ΩΩ is a bounded domain in RnRn. 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.