A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid

Abstract

We study the following initial-boundary value problem {ut − μt+αt(∂/∂t)∂2u/∂x2+(γ/x)(∂u/∂x) + fu = f1x,t, 1<x<R, t>0; u(1,t)=g1(t), u(R,t)=gR(t); u(x,0)=u~0(x)}, where γ>0,R>1 are given constants and f,f1,g1,gR,u~0,α, and μ are given functions. In Part 1, we use the Galerkin method and compactness method to prove the existence of a unique weak solution of the problem above on (0,T), for every T>0. In Part 2, we investigate asymptotic behavior of the solution as t→+∞. In Part 3, we prove the existence and uniqueness of a weak solution of problem {ut − μt+αt(∂/∂t)∂2u/∂x2+(γ/x)(∂u/∂x) + fu = f1x,t, 1<x<R, t>0; u(1,t)=g1(t), u(R,t)=gR(t)} associated with a “(η,T)-periodic condition” u(x,0)=ηu(x,T), where 0<η≤1 is given constant.