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−(μ(t)+α(t)(∂/∂t))(∂2u/∂x2+(γ/x)(∂u/∂x))+f(u)=f1(x, t), 1 < x < R, t > 0; u(1, t) = g1(t), u(R, t) = gR(t); u(x, 0) = ?̃?0(x)}, where γ > 0, R > 1 are given constants and f, f1, g1, gR, ?̃?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 − (μ(t) + α(t)(∂/∂t))(∂2u/∂x2 + (γ/x)(∂u/∂x)) + f(u) = f1(x, 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.

In the case of  = 1, () =  > 0, and () =  > 0 being the constants, the initial-boundary value problems (1)-( 3) are classical and have a long history of applications and mathematical development.We refer to the monographs of Al'shin et al. [1] and of Carroll and Showalter [2] for references and results on pseudoparabolic or Sobolev type equations.We also refer to [3] for asymptotic behavior and to [4] for nonlinear problems.Problems of this type arise in material science and physics, which have been extensively studied, and several results concerning existence, regularity, and asymptotic behavior have been established.
Equation (1) arises within frameworks of mathematical models in engineering and physical sciences (see [5][6][7][8][9][10][11] for references therein and interesting results on second grade fluids or a fourth grade fluid or other unsteady flows).It is well known that fluid solid mixtures are generally considered as second-grade fluids and are modeled as fluids with variable physical parameters; thus, an analysis is performed for a second-grade fluid with space dependent viscosity, elasticity, and density.
In [9], some unsteady flow problems of a second-grade fluid were considered.The flows are generated by the sudden application of a constant pressure gradient or by the impulsive motion of a boundary.Here, the velocities of the flows are described by the partial differential equations and exact analytic solutions of these differential equations are obtained.Suppose that the second-grade fluid is in a circular cylinder and is initially at rest, and the fluid starts suddenly due to the motion of the cylinder parallel to its length.The axis of the cylinder is chosen as the -axis.Using cylindrical polar coordinates, the governing partial differential equation is )  (, ) − , 0 <  < ,  > 0,  (, ) = ,  > 0,  (, 0) = 0, 0 ≤  < , where (, ) is the velocity along the -axis, ] is the kinematic viscosity,  is the material parameter, and  is the imposed magnetic field.In the boundary and initial conditions,  is the constant velocity at  =  and  is the radius of the cylinder.
In [6], two types of time-dependent flows were investigated.An eigenfunction expansion method was used to find the velocity distribution.The obtained solutions satisfy the boundary and initial conditions and the governing equation.Remarkably, some exact analytic solutions are possible for flows involving second-grade fluid with variable material properties in terms of trigonometric and Chebyshev functions.
In [5], Mahmood et al. have considered the longitudinal oscillatory motion of second-grade fluid between two infinite coaxial circular cylinders, oscillating along their common axis with given constant angular frequencies Ω 1 and Ω 2 .Velocity field and associated tangential stress of the motion were determined by using Laplace and Hankel transforms.In order to find exact analytic solutions for the flow of secondgrade fluid between two longitudinally oscillating cylinders, the following problem was studied: where 0 <  1 <  2 , , ,  1 ,  2 , Ω 1 , and Ω 2 are positive constants.The solutions obtained have been presented under series form in terms of Bessel functions  0 (),  0 (),  1 (),  1 (),  2 (), and  2 (), satisfying the governing equation and all imposed initial and boundary conditions.The nonlinear parabolic problems of the form (1)-( 3), with/without the term   + (/)  , were also studied in [12,13] and references therein.In [12], by using the Galerkin and compactness method in appropriate Sobolev spaces with weight, the authors proved the existence of a unique weak solution of the following initial and boundary value problem for nonlinear parabolic equation:   −  () (  +     ) +  (, ) =  (, ) , 0 <  < 1, 0 <  < ,         lim Furthermore, asymptotic behavior of the solution as  → +∞ was studied.In [13], the following nonlinear heat equation associated with Dirichlet-Robin conditions was investigated: (, ) ∈ (0, 1) × (0, ) ,   (0, ) = ℎ 0  (0, ) +  0 () , Condition (4), which we call "(, )-periodic condition," is known as a drifted periodic condition (see [14]).Indeed, if () = ( + ), ∀ ≥ 0, in the case of 0 < || ≤ 1, then we have which means with () = (1/ − 1)() satisfying the condition Note that (11) holds by the fact that With  = 1, (4) leads to -periodic condition and with  = −1, we have the antiperiodic condition The present paper is concerned with the second-grade fluid in a circular cylinder associated with the initial condition (3) or a drifted periodic condition (10).The extensive study of such flows is motivated by both their fundamental interest and their practical importance (see [9]).
This paper is a continuation of paper [15] dealing with the nonlinear pseudoparabolic equation (1) associated with the mixed inhomogeneous condition, in the case of  = 1, () =  > 0, () =  > 0 being the constants.It consists of five sections.First, preliminaries are done in Section 2. Under appropriate conditions, the existence of a unique weak solution of problems ( 1)-( 3) is proved in Section 3. Next, an asymptotic behavior of the solution of problems ( 1)-( 3), as  → +∞, is discussed in Section 4. Finally, Section 5 is devoted to the establishment, the existence, and uniqueness of a weak solution of problems ( 1), (2), and (4).
Because of mathematical context, the results obtained here generalize relatively the ones in [12,13,15], by improving the techniques used as before and with appropriate modifications.
On  1 , we shall use the following norm: We put 1 0 is a closed subspace of  1 and on  1 0 , two norms ‖V‖  1 and ‖V  ‖ are equivalent.
Note that  2 and  1 are also the Hilbert spaces with respect to the corresponding scalar products respectively.The norms in  2 and  1 induced by the corresponding scalar products are denoted by ‖ ⋅ ‖ 0 and ‖ ⋅ ‖ 1 , respectively. 1 0 is continuously and densely embedded in  2 .Identifying  2 with ( 2 )  (the dual of  2 ), we have  1 0 →  2 → ( 1 0 )  =  −1 ; on the other hand, the notation ⟨⋅, ⋅⟩ is used for the pairing between  1 0 and  −1 .We then have the following lemmas, the proofs of which can be found in [16].
Consider (⋅, ⋅) is the symmetric bilinear form on  1 0 × 1 0 defined by Then, the symmetric bilinear form (⋅, ⋅) is continuous on  1 0 ×  1 0 and coercive on  1 0 .We have also the following lemma.Lemma 5.There exists the Hilbert orthonormal base {  } of  2 consisting of the eigenfunctions   corresponding to the eigenvalue   such that Furthermore, the sequence {  /√  } is also the Hilbert orthonormal base of  1 0 with respect to the scalar product (⋅, ⋅).
On the other hand, we also have   satisfying the following boundary value problem: The proof of Lemma 5 can be found in [17, p. 87, Theorem 7.7], with  =  2 and  =  1 0 and (⋅, ⋅) as defined by (21).
Step 1 (the Faedo-Galerkin approximation (introduced by Lions [18])).Consider the basis {  } for  1 0 as in Lemma 5. We find the approximate solution of problem (24) in the form where the coefficients   satisfy the system of linear differential equations where The system of (30) can be rewritten in the form It is clear that for each  there exists a solution V  () in the form of (29) which satisfies (30) almost everywhere on 0 ≤  ≤ T for some T , 0 < T ≤ .The following estimates allow one to take T =  for all .
Step 2 (a priori estimates) (a) The First Estimate.Multiplying the th equation of (30) by   () and summing up with respect to , afterwards, integrating by parts with respect to the time variable from 0 to , we get after some rearrangements: By V 0 → Ṽ0 strongly in  1 0 , we have where  0 always indicates a bound depending on Ṽ0 .Put By the assumptions ( 3 )-( 6 ), we estimate without difficulty the following terms in (33) as follows: Hence, it follows from (33), (34), and (36) that where ∈  1 (0, ) .
(38) By Gronwall's lemma, we obtain from (37) that for all  ∈ N, for all , 0 ≤  ≤ T ≤ ; that is, T = , where   always indicates a bound depending on .
(b) The Second Estimate.Multiplying the th equation of (30) by 2 2    () and summing up with respect to , we have Integrating (40), we get We shall estimate the terms of (41) as follows: On the other hand, we have and hence (47) Step 3 (the limiting process).By (39), (46), and (47), we deduce that there exists a subsequence of {V  }, still denoted by {V  } such that Using a compactness lemma ( [18], Lions, p. 57), applied to (48), we can extract from the sequence {V  } a subsequence still denoted by {V  }, such that By the Riesz-Fischer theorem, we can extract from {V  } a subsequence still denoted by {V  }, such that where   is a constant independent of .
Lemma 7 is a slight improvement of a lemma used in [12] (or it can be found in Lions's book [18]).Now, we will prove the uniqueness of the solutions.Let V 1 and V 2 be two weak solutions of (24).Then, a weak solution of (55) with the right-hand side function replaced by f(, ) = −(V 1 + ) + (V 2 + ) and Ṽ0 = 0. Using Lemma 7, we have equality where By ( 6 ), we obtain It follows from ( 57)-( 59) that By Gronwall's lemma, V = 0. Assume now that ( 5 ) is replaced by  1 ∈  2 (  ); then we only have to show that {V   } is bounded in  2 (0, ;  1 0 ).Indeed, multiplying the th equation of (30) by    () and summing up with respect to , afterwards, integrating with respect to the time variable from 0 to , we get after some rearrangements where By the same estimates as above, we obtain . (64) Then, the sequence {V   } is bounded in  2 (0, ;  1 0 ).Applying a similar argument used as above, the limit V of the sequence {V  } in suitable function spaces is a unique weak solution of (24) satisfying (28).
First, we consider the following stationary problem: The weak solution of problem ( 68) is obtained from the following variational problem.Find  ∞ ∈  1 0 such that for all  ∈  1 0 , where (⋅, ⋅) is the symmetric bilinear form on  1 0 ×  1 0 defined by (21).We then have the following theorem.Proof.Consider the basis {  } for  1 0 as in Lemma 5. Put where   satisfies the following nonlinear equation system: By Brouwer's lemma (see Lions [18], Lemma 4.3, p. 53), it follows from the hypotheses (  4 )-(  6 ) that systems (70) and (71) have a solution   .
Multiplying the th equation of system (71) by   , and then summing up with respect to , we have By using ( 6 ), we obtain Hence, we deduce from (74) that and D1 is a constant independent of .
By means of (75) and Lemma 3, the sequence {  } has a subsequence still denoted by {  } such that On the other hand, by (76) 2 and the continuity of , we have Passing to the limit in (71), we find without difficulty from (76) and (77) that  ∞ satisfies the equation Equation ( 78) holds for every  = 1, 2, . ..; that is, (69) is true.
The solution of problem ( 69) is unique, which can be shown by the same arguments as in the proof of Theorem 6.

Now we consider asymptotic behavior of the solution 𝑢(𝑡)
as  → +∞.
Then, we have the following theorem.
Theorem 11.Let  > 0 and ( 2 )-( 6 ) hold.Then, problem (112) has a unique weak solution V such that Proof.The proof consists of several steps.
Find a function V  () in the form (29) satisfying the nonlinear differential equation system (30) 1 and the (, )periodic condition: We consider an initial value problem given by (30), where V 0 is given in   .
It is clear that, for each , there exists a solution V  () in the form (29) which satisfies (30) almost everywhere on 0 ≤  ≤ T for some T , 0 < T ≤ .The following a priori estimates allow us to take T =  for all .
On the other hand, multiplying the th equation of (30) 1 by    () and summing up with respect to , afterwards, integrating with respect to the time variable from 0 to , we get after some rearrangements where Moreover, for all  ∈ N, for all  ∈ [0, ], where   always indicates a bound depending on .
For ,  fixed, passing to the limit in (154) by ( 140) and (153), we get is fulfilled.