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

الملخص EN

We study the following initial-boundary value problem { u t − μ t + α t ( ∂ / ∂ t ) ∂ 2 u / ∂ x 2 + ( γ / x ) ( ∂ u / ∂ x ) + f u = f 1 x , t , 1 < x < R , t > 0 ; u ( 1 , t ) = g 1 ( t ) , u ( R , t ) = g R ( t ) ; u ( x , 0 ) = u ~ 0 ( x ) } , where γ > 0 , R > 1 are given constants and f , f 1 , g 1 , g R , 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 { u t − μ t + α t ( ∂ / ∂ t ) ∂ 2 u / ∂ x 2 + ( γ / x ) ( ∂ u / ∂ x ) + f u = f 1 x , t , 1 < x < R , t > 0 ; u ( 1 , t ) = g 1 ( t ) , u ( R , t ) = g R ( t ) } associated with a “ ( η , T ) -periodic condition” u ( x , 0 ) = η u ( x , T ) , where 0 < η ≤ 1 is given constant.