A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid
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 an...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2016-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2016/3875324 |
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| Summary: | 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. |
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| ISSN: | 1026-0226 1607-887X |