A Nonhomogeneous Dirichlet Problem for a Nonlinear

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2016, Article ID 3875324, 17 pages http://dx.doi.org/10.1155/2016/3875324

Research Article A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid Le Thi Phuong Ngoc,1 Truong Thi Nhan,2,3 and Nguyen Thanh Long3 1

University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang, Vietnam The Faculty of Natural Basic Sciences, Vietnamese Naval Academy, 30 Tran Phu Street, Nha Trang, Vietnam 3 Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam 2

Correspondence should be addressed to Nguyen Thanh Long; [email protected] Received 18 September 2016; Accepted 13 November 2016 Academic Editor: Andrew Pickering Copyright © 2016 Le Thi Phuong Ngoc et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the following initial-boundary value problem {𝑢𝑡 −(𝜇(𝑡)+𝛼(𝑡)(𝜕/𝜕𝑡))(𝜕2 𝑢/𝜕𝑥2 +(𝛾/𝑥)(𝜕𝑢/𝜕𝑥)) + 𝑓(𝑢) = 𝑓1 (𝑥, 𝑡), 1 < 𝑥 < 𝑅, ̃ 0 (𝑥)}, where 𝛾 > 0, 𝑅 > 1 are given constants and 𝑓, 𝑓1 , 𝑔1 , 𝑔𝑅 , 𝑢 ̃ 0 , 𝛼, and 𝜇 are 𝑡 > 0; 𝑢(1, 𝑡) = 𝑔1 (𝑡), 𝑢(𝑅, 𝑡) = 𝑔𝑅 (𝑡); 𝑢(𝑥, 0) = 𝑢 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))(𝜕2 𝑢/𝜕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.

1. Introduction In this paper, we consider the following nonlinear pseudoparabolic equation: 𝑢𝑡 − (𝜇 (𝑡) + 𝛼 (𝑡) = 𝑓1 (𝑥, 𝑡) ,

𝜕 𝜕2 𝑢 𝛾 𝜕𝑢 )( 2 + ) + 𝑓 (𝑢) 𝜕𝑡 𝜕𝑥 𝑥 𝜕𝑥

(1)

1 < 𝑥 < 𝑅, 𝑡 > 0,

associated with the boundary conditions 𝑢 (1, 𝑡) = 𝑔1 (𝑡) , 𝑢 (𝑅, 𝑡) = 𝑔𝑅 (𝑡)

(2)

and the initial condition ̃ 0 (𝑥) , 𝑢 (𝑥, 0) = 𝑢

(3)

or the “(𝜂, 𝑇)-periodic condition” 𝑢 (𝑥, 0) = 𝜂𝑢 (𝑥, 𝑇) ,

(4)

where 𝛾 > 0, 𝑅 > 1, and 0 < |𝜂| ≤ 1 are given constants ̃ 0 , 𝛼, and 𝜇 are given functions satisfying and 𝑓, 𝑓1 , 𝑔1 , 𝑔𝑅 , 𝑢 conditions specified later. In the case of 𝛾 = 1, 𝜇(𝑡) = 𝜇 > 0, and 𝛼(𝑡) = 𝛼 > 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 [5–11] 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

2

Discrete Dynamics in Nature and Society

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 𝑧-axis. Using cylindrical polar coordinates, the governing partial differential equation is

series form in terms of Bessel functions 𝐽0 (𝑥), 𝑌0 (𝑥), 𝐽1 (𝑥), 𝑌1 (𝑥), 𝐽2 (𝑥), and 𝑌2 (𝑥), satisfying the governing equation and all imposed initial and boundary conditions. The nonlinear parabolic problems of the form (1)–(3), with/without the term 𝑢𝑟𝑟 + (𝛾/𝑟)𝑢𝑟 , 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: 𝛾 𝑢𝑡 − 𝑎 (𝑡) (𝑢𝑟𝑟 + 𝑢𝑟 ) + 𝐹 (𝑟, 𝑢) = 𝑓 (𝑟, 𝑡) , 𝑟 0 < 𝑟 < 1, 0 < 𝑡 < 𝑇, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 𝛾/2 󵄨󵄨 lim 𝑟 𝑢𝑟 (𝑟, 𝑡)󵄨󵄨󵄨 < +∞, 󵄨󵄨𝑟→0+ 󵄨󵄨

2

𝜕𝑤 1 𝜕 𝜕 𝜕 = (] + 𝛼 ) ( 2 + ) 𝑤 (𝑟, 𝑡) − 𝑁𝑤, 𝜕𝑡 𝜕𝑡 𝜕𝑟 𝑟 𝜕𝑟 0 < 𝑟 < 𝑎, 𝑡 > 0, (5) 𝑤 (𝑎, 𝑡) = 𝑊, 𝑤 (𝑟, 0) = 0,

𝑡 > 0, 0 ≤ 𝑟 < 𝑎,

where 𝑤(𝑟, 𝑡) is the velocity along the 𝑧-axis, ] is the kinematic viscosity, 𝛼 is the material parameter, and 𝑁 is the imposed magnetic field. In the boundary and initial conditions, 𝑊 is the constant velocity at 𝑟 = 𝑎 and 𝑎 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 Ω1 and Ω2 . 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:

𝑢𝑟 (1, 𝑡) + ℎ (𝑡) (𝑢 (1, 𝑡) − 𝑢0 ) = 0, 𝑢 (𝑟, 0) = 𝑢0 (𝑟) . Furthermore, asymptotic behavior of the solution as 𝑡 → +∞ was studied. In [13], the following nonlinear heat equation associated with Dirichlet-Robin conditions was investigated: 𝑢𝑡 −

𝜕 [𝜇 (𝑥, 𝑡) 𝑢𝑥 ] + 𝑓 (𝑢) = 𝑓1 (𝑥, 𝑡) , 𝜕𝑥 (𝑥, 𝑡) ∈ (0, 1) × (0, 𝑇) , 𝑢𝑥 (0, 𝑡) = ℎ0 𝑢 (0, 𝑡) + 𝑔0 (𝑡) ,

(8)

−𝑢𝑥 (1, 𝑡) = ℎ1 𝑢 (1, 𝑡) + 𝑔1 (𝑡) , 𝑢 (𝑥, 0) = 𝑢0 (𝑥) .

Condition (4), which we call “(𝜂, 𝑇)-periodic condition,” is known as a drifted periodic condition (see [14]). Indeed, if 𝑢(𝑡) = 𝜂𝑢(𝑡 + 𝑇), ∀𝑡 ≥ 0, in the case of 0 < |𝜂| ≤ 1, then we have 𝑢 (𝑡 + 𝑇) =

1 1 𝑢 (𝑡) = 𝑢 (𝑡) + ( − 1) 𝑢 (𝑡) , ∀𝑡 ≥ 0, 𝜂 𝜂

(9)

which means 𝑢 (𝑡 + 𝑇) = 𝑢 (𝑡) + 𝛿 (𝑡) , ∀𝑡 ≥ 0,

(10)

with 𝛿(𝑡) = (1/𝜂 − 1)𝑢(𝑡) satisfying the condition

𝜕V 1 𝜕 𝜕 𝜕2 = (𝜇 + 𝛼 ) ( 2 + ) V (𝑟, 𝑡) , 𝜕𝑡 𝜕𝑡 𝜕𝑟 𝑟 𝜕𝑟

𝛿 (𝑡) = 𝜂𝛿 (𝑡 + 𝑇) , ∀𝑡 ≥ 0.

𝑅1 < 𝑟 < 𝑅2 , 𝑡 > 0, V (𝑅1 , 𝑡) = 𝑉1 sin (Ω1 𝑡) ,

(7)

(11)

Note that (11) holds by the fact that (6)

V (𝑅2 , 𝑡) = 𝑉2 sin (Ω2 𝑡) , V (𝑟, 0) = 0, 𝑅1 ≤ 𝑟 ≤ 𝑅2 , where 0 < 𝑅1 < 𝑅2 , 𝜇, 𝛼, 𝑉1 , 𝑉2 , Ω1 , and Ω2 are positive constants. The solutions obtained have been presented under

1 1 𝜂𝛿 (𝑡 + 𝑇) = 𝜂 [( − 1) 𝑢 (𝑡 + 𝑇)] = ( − 1) 𝑢 (𝑡) 𝜂 𝜂

(12)

= 𝛿 (𝑡) , ∀𝑡 ≥ 0. With 𝜂 = 1, (4) leads to 𝑇-periodic condition 𝑢 (𝑥, 0) = 𝑢 (𝑥, 𝑇) ,

(13)


3 Note that 𝐿2 and 𝐻1 are also the Hilbert spaces with respect to the corresponding scalar products

and with 𝜂 = −1, we have the antiperiodic condition 𝑢 (𝑥, 0) = −𝑢 (𝑥, 𝑇) .

(14)

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 𝛾 = 1, 𝜇(𝑡) = 𝜇 > 0, 𝛼(𝑡) = 𝛼 > 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 𝑡 → +∞, 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.

2. Preliminaries Put Ω = (1, 𝑅), 𝑄𝑇 = Ω × (0, 𝑇), 𝑇 > 0. We omit the definitions of the usual function spaces: 𝐶𝑚 (Ω), 𝐿𝑝 (Ω), and 𝑊𝑚,𝑝 (Ω). We define 𝑊𝑚,𝑝 = 𝑊𝑚,𝑝 (Ω), 𝐿𝑝 = 𝑊0,𝑝 (Ω) and 𝐻𝑚 = 𝑊𝑚,2 (Ω), 1 ≤ 𝑝 ≤ ∞, 𝑚 = 0, 1, . . .. The norm in 𝐿2 is denoted by ‖ ⋅ ‖. We also denote by (⋅, ⋅) the scalar product in 𝐿2 . We denote by ‖ ⋅ ‖𝑋 the norm of a Banach space 𝑋 and by 𝑋󸀠 the dual space of 𝑋. We denote by 𝐿𝑝 (0, 𝑇; 𝑋)1 ≤ 𝑝 ≤ ∞ for the Banach space of the real functions 𝑢 : (0, 𝑇) → 𝑋 measurable, such that ‖𝑢‖𝐿𝑝 (0,𝑇;𝑋) = (∫

𝑇

0

𝑝 ‖𝑢 (𝑡)‖𝑋 𝑑𝑡)

0 0 and (𝐻1 )–(𝐻6 ) hold. Then, problem (24) has a unique weak solution V such that V ∈ 𝐿∞ (0, 𝑇; 𝐻01 ) ,

(𝐻5 ) 𝑓1 ∈ 𝐿1 (0, 𝑇; 𝐿2 ). (𝐻6 ) 𝑓 ∈ 𝐶0 (R; R) satisfies the condition that there exists positive constant 𝛿 such that (𝑦 − 𝑧)(𝑓(𝑦) − 𝑓(𝑧)) ≥ −𝛿|𝑦 − 𝑧|2 , for all 𝑦, 𝑧 ∈ R. In case 𝑔1 ≠ 0 or 𝑔𝑅 ≠ 0, it is clearly that problems (1)–(3) reduce to a problem with homogeneous boundary conditions by the suitable transformation. Indeed, putting 𝜑(𝑥, 𝑡) = ((𝑥 − 1)/(𝑅 − 1))𝑔𝑅 (𝑡) + ((𝑅 − 𝑥)/(𝑅 − 1))𝑔1 (𝑡), by the transformation V(𝑥, 𝑡) = 𝑢(𝑥, 𝑡)−𝜑(𝑥, 𝑡), problems (1)–(3) reduce to the following problem: 𝜕 𝜕2 V 𝛾 𝜕V V𝑡 − (𝜇 (𝑡) + 𝛼 (𝑡) ) ( 2 + ) + 𝑓 (V + 𝜑) 𝜕𝑡 𝜕𝑥 𝑥 𝜕𝑥 = 𝑓1 (𝑥, 𝑡) , 1 < 𝑥 < 𝑅, 𝑡 > 0,

(24)

(27)

𝑡V𝑡 ∈ 𝐿2 (0, 𝑇; 𝐻01 ) .

Moreover, if (𝐻5 ) is replaced by 𝑓1 ∈ 𝐿2 (𝑄𝑇 ), then the solution V satisfies V ∈ 𝐿∞ (0, 𝑇; 𝐻01 ) ,

(28)

V𝑡 ∈ 𝐿2 (0, 𝑇; 𝐻01 ) . Proof. The proof consists of several steps.

Step 1 (the Faedo-Galerkin approximation (introduced by Lions [18])). Consider the basis {𝑤𝑗 } for 𝐻01 as in Lemma 5. We find the approximate solution of problem (24) in the form 𝑚

V (1, 𝑡) = V (𝑅, 𝑡) = 0,

V𝑚 (𝑡) = ∑ 𝑐𝑚𝑗 (𝑡) 𝑤𝑗 ,

V (𝑥, 0) = ̃V0 (𝑥) ,

(29)

𝑗=1

where the coefficients 𝑐𝑚𝑗 satisfy the system of linear differential equations

where 𝑓1 (𝑥, 𝑡) = 𝑓1 (𝑥, 𝑡) −

(26)

+ ⟨𝑓 (V (𝑡) + 𝜑 (𝑡)) , 𝑤⟩ = ⟨𝑓1 (𝑡) , 𝑤⟩ ,

1 [(𝑥 − 1) 𝑔𝑅󸀠 (𝑡) 𝑅−1

󸀠 󸀠 ⟨V𝑚 (𝑡) , 𝑤𝑗 ⟩ + 𝛼 (𝑡) 𝑎 (V𝑚 (𝑡) , 𝑤𝑗 )

+ (𝑅 − 𝑥) 𝑔1󸀠 (𝑡)] 𝛾 + [𝜇 (𝑡) (𝑔𝑅 (𝑡) − 𝑔1 (𝑡)) (𝑅 − 1) 𝑥

+ 𝜇 (𝑡) 𝑎 (V𝑚 (𝑡) , 𝑤𝑗 ) (25)

+ ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) , 𝑤𝑗 ⟩ = ⟨𝑓1 (𝑡) , 𝑤𝑗 ⟩ ,

+ 𝛼 (𝑡) (𝑔𝑅󸀠 (𝑡) − 𝑔1󸀠 (𝑡))] , ̃ 0 (𝑥) − 𝜑 (𝑥, 0) , ̃V0 ∈ 𝐻01 ̃V0 (𝑥) = 𝑢 ̃ 0 , 𝑔1 , and 𝑔𝑅 satisfy the condition 𝑢 ̃ 0 (1)−𝑔1 (0) = 𝑢 ̃ 0 (𝑅)− and 𝑢 𝑔𝑅 (0) = 0. The weak formulation of the initial-boundary value problem (24) can be given in the following manner: Find

(30)

1 ≤ 𝑗 ≤ 𝑚, V𝑚 (0) = V0𝑚 , where 𝑚

V0𝑚 = ∑ 𝛼𝑚𝑗 𝑤𝑗 󳨀→ ̃V0 𝑗=1

strongly in 𝐻01 .

(31)


5 𝑡

The system of (30) can be rewritten in the form

− 2 ∫ ⟨𝑓 (V𝑚 (𝑠) + 𝜑 (𝑠)) , V𝑚 (𝑠)⟩ 𝑑𝑠 0

󸀠 𝑐𝑚𝑗 (𝑡) +

+ =

𝜆𝑗 𝜇 (𝑡) 1 + 𝜆𝑗 𝛼 (𝑡) 1

1 + 𝜆𝑗 𝛼 (𝑡) 1 1 + 𝜆𝑗 𝛼 (𝑡)

𝑐𝑚𝑗 (0) = 𝛼𝑚𝑗 ,

𝑡

󵄩2 󵄩 ≤ 2𝛿 ∫ 󵄩󵄩󵄩V𝑚 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0

𝑐𝑚𝑗 (𝑡)

𝑡

⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) , 𝑤𝑗 ⟩

󵄩 󵄩 󵄩 󵄩 + 2 ∫ 󵄩󵄩󵄩𝑓 (𝜑 (𝑠))󵄩󵄩󵄩0 󵄩󵄩󵄩V𝑚 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0

(32)

𝑡

𝑇

󵄩2 󵄩 󵄩 󵄩2 ≤ (2𝛿 + 1) ∫ 󵄩󵄩󵄩V𝑚 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 + ∫ 󵄩󵄩󵄩𝑓 (𝜑 (𝑠))󵄩󵄩󵄩0 𝑑𝑠; 0 0

⟨𝑓1 (𝑡) , 𝑤𝑗 ⟩ ,

𝑡

2 ∫ ⟨𝑓1 (𝑠) , V𝑚 (𝑠)⟩ 𝑑𝑠

1 ≤ 𝑗 ≤ 𝑚.

0

𝑡 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩2 ≤ 󵄩󵄩󵄩󵄩𝑓1 󵄩󵄩󵄩󵄩𝐿1 (0,𝑇;𝐿2 ) + ∫ 󵄩󵄩󵄩󵄩𝑓1 (𝑠)󵄩󵄩󵄩󵄩0 󵄩󵄩󵄩V𝑚 (𝑠)󵄩󵄩󵄩0 𝑑𝑠

It is clear that for each 𝑚 there exists a solution V𝑚 (𝑡) in the form of (29) which satisfies (30) almost everywhere on ̃ 𝑚, 0 < 𝑇 ̃ 𝑚 ≤ 𝑇. The following estimates ̃ 𝑚 for some 𝑇 0≤𝑡≤𝑇 ̃ allow one to take 𝑇𝑚 = 𝑇 for all 𝑚.

0

󵄩 󵄩 ≤ 󵄩󵄩󵄩󵄩𝑓1 󵄩󵄩󵄩󵄩𝐿1 (0,𝑇;𝐿2 ) + ∫

𝑡

0

󵄩󵄩 󵄩 󵄩󵄩𝑓1 (𝑠)󵄩󵄩󵄩 𝑆𝑚 (𝑠) 𝑑𝑠. 󵄩 󵄩0 (36)

Step 2 (a priori estimates) (a) The First Estimate. Multiplying the 𝑗th equation of (30) by 𝑐𝑚𝑗 (𝑡) and summing up with respect to 𝑗, afterwards, integrating by parts with respect to the time variable from 0 to 𝑡, we get after some rearrangements: 󵄩󵄩 󵄩2 󵄩2 󵄩 󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼 (𝑡) 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 󵄩2 󵄩 󵄩2 󵄩 = 󵄩󵄩󵄩V0𝑚 󵄩󵄩󵄩0 + 𝛼 (0) 󵄩󵄩󵄩V0𝑚𝑥 󵄩󵄩󵄩0 𝑡

󵄩 󵄩2 − ∫ (2𝜇 (𝑠) − 𝛼 (𝑠)) 󵄩󵄩󵄩V𝑚𝑥 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0

Hence, it follows from (33), (34), and (36) that 𝑡

𝑆𝑚 (𝑡) ≤ 𝐶𝑇(1) + ∫ 𝑑𝑇(1) (𝑠) 𝑆𝑚 (𝑠) 𝑑𝑠, 0

(37)

where 𝑇 󵄩 󵄩 󵄩 󵄩2 𝐶𝑇(1) = 𝑆0 + 󵄩󵄩󵄩󵄩𝑓1 󵄩󵄩󵄩󵄩𝐿1 (0,𝑇;𝐿2 ) + ∫ 󵄩󵄩󵄩𝑓 (𝜑 (𝑠))󵄩󵄩󵄩0 𝑑𝑠, 0

󸀠

(33)

𝑡

1 󵄨󵄨 󵄩 󵄩 󵄨 󵄨󵄨2𝜇 (𝑠) − 𝛼󸀠 (𝑠)󵄨󵄨󵄨 , 𝑑𝑇(1) (𝑠) = 1 + 2𝛿 + 󵄩󵄩󵄩󵄩𝑓1 (𝑠)󵄩󵄩󵄩󵄩0 + 󵄨 𝛼∗ 󵄨

− 2 ∫ ⟨𝑓 (V𝑚 (𝑠) + 𝜑 (𝑠)) , V𝑚 (𝑠)⟩ 𝑑𝑠

(38)

𝑑𝑇(1) ∈ 𝐿1 (0, 𝑇) .

0

𝑡

By Gronwall’s lemma, we obtain from (37) that

+ 2 ∫ ⟨𝑓1 (𝑠) , V𝑚 (𝑠)⟩ 𝑑𝑠. 0

𝑡

By V0𝑚 → ̃V0 strongly in

𝐻01 ,

𝑆𝑚 (𝑡) ≤ 𝐶𝑇(1) exp (∫ 𝑑𝑇(1) (𝑠) 𝑑𝑠) ≤ 𝐶𝑇 ,

we have

󵄩󵄩 󵄩󵄩2 󵄩2 󵄩 󵄩󵄩V0𝑚 󵄩󵄩0 + 𝛼 (0) 󵄩󵄩󵄩V0𝑚𝑥 󵄩󵄩󵄩0 ≤ 𝑆0 , ∀𝑚,

0

(34)

where 𝑆0 always indicates a bound depending on ̃V0 . Put 󵄩 󵄩2 󵄩2 󵄩 𝑆𝑚 (𝑡) = 󵄩󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼∗ 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 .

󵄩 󵄩2 − ∫ (2𝜇 (𝑠) − 𝛼 (𝑠)) 󵄩󵄩󵄩V𝑚𝑥 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0 ≤

󸀠

𝑡

1 󵄨 󵄨 ∫ 󵄨󵄨󵄨2𝜇 (𝑠) − 𝛼󸀠 (𝑠)󵄨󵄨󵄨󵄨 𝑆𝑚 (𝑠) 𝑑𝑠; 𝛼∗ 0 󵄨

̃ 𝑚 ≤ 𝑇; that is, 𝑇 ̃ 𝑚 = 𝑇, where for all 𝑚 ∈ N, for all 𝑡, 0 ≤ 𝑡 ≤ 𝑇 𝐶𝑇 always indicates a bound depending on 𝑇. (b) The Second Estimate. Multiplying the 𝑗th equation of (30) 󸀠 by 2𝑡2 𝑐𝑚𝑗 (𝑡) and summing up with respect to 𝑗, we have

(35)

By the assumptions (𝐻3 )–(𝐻6 ), we estimate without difficulty the following terms in (33) as follows: 𝑡

(39)

󵄩 󸀠 󵄩2 󵄩2 󵄩 󸀠 2 󵄩󵄩󵄩󵄩𝑡V𝑚 (𝑡)󵄩󵄩󵄩󵄩0 + 2𝛼 (𝑡) 󵄩󵄩󵄩󵄩𝑡V𝑚𝑥 (𝑡)󵄩󵄩󵄩󵄩0 +

𝑑 󵄩2 󵄩 [𝜇 (𝑡) 󵄩󵄩󵄩𝑡V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ] 𝑑𝑡

󸀠󵄩 󵄩2 = (𝑡2 𝜇 (𝑡)) 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 󸀠 − 2𝑡2 ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) , V𝑚 (𝑡)⟩ 󸀠 + 2𝑡2 ⟨𝑓1 (𝑡) , V𝑚 (𝑡)⟩ .

(40)

6

Discrete Dynamics in Nature and Society Integrating (40), we get

≤ 2𝑇2 (

𝑇

𝑡 𝑡 󵄩 󸀠 󵄩2 󵄩2 󵄩 󸀠 2 ∫ 󵄩󵄩󵄩󵄩𝑠V𝑚 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠 + 2 ∫ 𝛼 (𝑠) 󵄩󵄩󵄩󵄩𝑠V𝑚𝑥 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠 0 0

+

󵄩2 󵄩 + 𝜇 (𝑡) 󵄩󵄩󵄩𝑡V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 󸀠󵄩 󵄩2 = ∫ (𝑠2 𝜇 (𝑠)) 󵄩󵄩󵄩V𝑚𝑥 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0

(41)

𝑡

󸀠 − 2 ∫ ⟨𝑠𝑓 (V𝑚 (𝑠) + 𝜑 (𝑠)) , 𝑠V𝑚 (𝑠)⟩ 𝑑𝑠 0

󸀠 + 2 ∫ ⟨𝑠𝑓1 (𝑠) , 𝑠V𝑚 (𝑠)⟩ 𝑑𝑠. 0

󵄩 󵄩2 + 2 ∫ 󵄩󵄩󵄩󵄩𝑠𝑓1 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠 0 + 2𝑇2 (

󵄩2 󵄩 ∫ (𝑠 𝜇 (𝑠)) 󵄩󵄩󵄩V𝑚𝑥 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0 󸀠

󸀠 󵄨󵄨 1 𝑡 󵄨󵄨󵄨 2 ∫ 󵄨󵄨(𝑠 𝜇 (𝑠)) 󵄨󵄨󵄨 𝑆𝑚 (𝑠) 𝑑𝑠 󵄨 𝛼∗ 0 󵄨

≤

󸀠 󵄨󵄨 𝐶𝑇 𝑇 󵄨󵄨󵄨 2 ∫ 󵄨(𝑠 𝜇 (𝑠)) 󵄨󵄨󵄨 𝑑𝑠; 󵄨 𝛼∗ 0 󵄨󵄨

𝑡 𝑡 󵄩 󸀠 󵄩 󸀠 󵄩2 󵄩2 ∫ 󵄩󵄩󵄩󵄩𝑠V𝑚 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠 + 2𝛼∗ ∫ 󵄩󵄩󵄩󵄩𝑠V𝑚𝑥 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠 0 0

𝑇

We shall estimate the terms of (41) as follows:

≤

It follows from (41)–(43) and (45) that

𝑇󵄨 󸀠 󵄨󵄨 󵄩2 𝐶 󵄩 󵄨 + 𝜇∗ 󵄩󵄩󵄩𝑡V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ≤ 𝑇 ∫ 󵄨󵄨󵄨(𝑠2 𝜇 (𝑠)) 󵄨󵄨󵄨 𝑑𝑠 󵄨 𝛼∗ 0 󵄨

𝑡

2

1 𝑡 󵄩󵄩 󸀠 󵄩2 ∫ 󵄩󵄩𝑠V (𝑠)󵄩󵄩󵄩 𝑑𝑠. 2 0 󵄩 𝑚 󵄩0 (45)

𝑡

𝑡

𝑅𝛾+1 − 1 ) sup 𝑓2 (𝑧) 𝑑𝑥 𝛾+1 |𝑧|≤𝐶

𝑅𝛾+1 − 1 ) sup 𝑓2 (𝑧) 𝑑𝑥 ≤ 𝐶𝑇 , 𝛾+1 |𝑧|≤𝐶 𝑇

(42)

for all 𝑚 ∈ N, for all 𝑡 ∈ [0, 𝑇], where 𝐶𝑇 always indicates a bound depending on 𝑇. 󸀠 By (𝑡V𝑚𝑥 )󸀠 = 𝑡V𝑚𝑥 + V𝑚𝑥 and (39) and (46), we deduce that 󵄩 󵄩 󵄩󵄩󵄩(𝑡V )󸀠 󵄩󵄩󵄩 󵄩󵄩 󸀠 󵄩󵄩 󵄩󵄩 𝑚𝑥 󵄩󵄩𝐿2 (𝑄𝑇 ) ≤ 󵄩󵄩󵄩𝑡V𝑚𝑥 󵄩󵄩󵄩𝐿2 (𝑄𝑇 ) + 󵄩󵄩󵄩V𝑚𝑥 󵄩󵄩󵄩𝐿2 (𝑄𝑇 )

𝑡

𝑇

󸀠 2 ∫ ⟨𝑠𝑓1 (𝑠) , 𝑠V𝑚 (𝑠)⟩ 𝑑𝑠 0

𝑇

1 𝑡󵄩 󸀠 󵄩 󵄩2 󵄩2 ≤ 2 ∫ 󵄩󵄩󵄩󵄩𝑠𝑓1 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠 + ∫ 󵄩󵄩󵄩󵄩𝑠V𝑚 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠. 2 0 0

󵄩 󵄩 + 󵄩󵄩󵄩𝑔𝑅 󵄩󵄩󵄩𝐶0 ([0,𝑇]) ≡ 𝐶𝑇 ,

󵄩 󵄩 + √𝑇 󵄩󵄩󵄩V𝑚 󵄩󵄩󵄩𝐿2 (0,𝑇;𝐻1 ) ≤ 𝐶𝑇 . Step 3 (the limiting process). By (39), (46), and (47), we deduce that there exists a subsequence of {V𝑚 }, still denoted by {V𝑚 } such that V𝑚 󳨀→ V

(44)

in 𝐿∞ (0, 𝑇; 𝐻01 ) weakly∗ ,

󸀠

(𝑡V𝑚 ) 󳨀→ (𝑡V)󸀠

2 ∫ ⟨𝑠𝑓 (V𝑚 (𝑠) + 0

󸀠 𝜑 (𝑠)) , 𝑠V𝑚

(𝑠)⟩ 𝑑𝑠

𝑡

󵄩2 󵄩 ≤ 2 ∫ 󵄩󵄩󵄩𝑠𝑓 (V𝑚 (𝑠) + 𝜑 (𝑠))󵄩󵄩󵄩0 𝑑𝑠 0 +

1 𝑡 󵄩󵄩 󸀠 󵄩2 ∫ 󵄩󵄩𝑠V (𝑠)󵄩󵄩󵄩 𝑑𝑠 2 0 󵄩 𝑚 󵄩0 𝑡

𝑅

0

1

≤ 2 ∫ 𝑠2 𝑑𝑠 ∫ 𝑥𝛾 sup 𝑓2 (𝑧) 𝑑𝑥 |𝑧|≤𝐶𝑇

1 𝑡󵄩 󸀠 󵄩2 + ∫ 󵄩󵄩󵄩󵄩𝑠V𝑚 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠 2 0

in 𝐿2 (0, 𝑇; 𝐻01 ) weakly.

(48)

Using a compactness lemma ([18], Lions, p. 57), applied to (48), we can extract from the sequence {V𝑚 } a subsequence still denoted by {V𝑚 }, such that 𝑡V𝑚 󳨀→ 𝑡V

and hence

(47)

0

󵄨 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄨󵄨 󵄨󵄨V𝑚 (𝑥, 𝑠)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜑 (𝑥, 𝑠)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩V𝑚 (𝑠)󵄩󵄩󵄩𝐶0 (Ω) + 󵄨󵄨󵄨𝑔1 (𝑠)󵄨󵄨󵄨 󵄨 󵄨 + 󵄨󵄨󵄨𝑔𝑅 (𝑠)󵄨󵄨󵄨 (𝑅 − 1) 𝐶𝑇 󵄩󵄩 󵄩󵄩 ≤√ + 󵄩󵄩𝑔1 󵄩󵄩𝐶0 ([0,𝑇]) 𝛼∗

󵄩2 󵄩 󸀠 ≤ √ ∫ 󵄩󵄩󵄩𝑠V𝑚𝑥 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0

(43)

On the other hand, we have

𝑡

(46)

strongly in 𝐿2 (𝑄𝑇 ) .

(49)

By the Riesz-Fischer theorem, we can extract from {V𝑚 } a subsequence still denoted by {V𝑚 }, such that V𝑚 (𝑥, 𝑡) 󳨀→ V (𝑥, 𝑡)

a.e. (𝑥, 𝑡) in 𝑄𝑇 .

(50)

Because 𝑓 is continuous, it gives 𝑓 (V𝑚 (𝑥, 𝑡) + 𝜑 (𝑥, 𝑡)) 󳨀→ 𝑓 (V (𝑥, 𝑡) + 𝜑 (𝑥, 𝑡)) a.e. (𝑥, 𝑡) in 𝑄𝑇 . On the other hand, by (𝐻6 ), it follows from (44) that 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨𝑓 (V𝑚 (𝑥, 𝑡) + 𝜑 (𝑥, 𝑡))󵄨󵄨󵄨 ≤ sup 󵄨󵄨󵄨𝑓 (𝑧)󵄨󵄨󵄨 ≤ 𝐶𝑇 , |𝑧|≤𝐶𝑇

where 𝐶𝑇 is a constant independent of 𝑚.

(51)

(52)


7

Using the dominated convergence theorem, (51) and (52) yield 𝑓 (V𝑚 + 𝜑) 󳨀→ 𝑓 (V + 𝜑)

2

strongly in 𝐿 (𝑄𝑇 ) .

By (𝐻6 ), we obtain 𝑡

2 ∫ ⟨𝑓 (V1 + 𝜑) − 𝑓 (V2 + 𝜑) , V (𝑠)⟩ 𝑑𝑠

(53)

0

Passing to the limit in (30) by (31), (48), and (53), we obtain 𝑑 [⟨V (𝑡) , 𝑤⟩ + 𝛼 (𝑡) 𝑎 (V (𝑡) , 𝑤)] 𝑑𝑡

0

0

󵄩 󵄩2 − ∫ (2𝜇 (𝑠) − 𝛼 (𝑠)) 󵄩󵄩󵄩V𝑥 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0 1 𝑡 󵄨󵄨 󵄨 ∫ 󵄨󵄨2𝜇 (𝑠) − 𝛼󸀠 (𝑠)󵄨󵄨󵄨󵄨 𝜎1 (𝑠) 𝑑𝑠. 𝛼∗ 0 󵄨

It follows from (57)–(59) that

∀𝑤 ∈ 𝐻01 , a.e., 𝑡 ∈ (0, 𝑇) ,

𝑡

V (0) = ̃V0 .

𝜎1 (𝑡) ≤ ∫ (2𝛿 + 0

Step 4 (uniqueness of the solution). First, we shall need the following lemma. Lemma 7. Let V be the weak solution of the following problem: 𝜕 𝜕2 V 𝛾 𝜕V ̃ (𝑥, 𝑡) , )( 2 + )=𝑓 𝜕𝑡 𝜕𝑥 𝑥 𝜕𝑥

V𝑡 − (𝜇 (𝑡) + 𝛼 (𝑡)

1 < 𝑥 < 𝑅, 0 < 𝑡 < 𝑇, V (1, 𝑡) = V (𝑅, 𝑡) = 0,

(55)

2

𝑡V𝑡 ∈ 𝐿

(0, 𝑇; 𝐻01 ) ,

1 𝛼∗

󵄨󵄨 󵄨 󵄨󵄨2𝜇 (𝑠) − 𝛼󸀠 (𝑠)󵄨󵄨󵄨) 𝜎1 (𝑠) 𝑑𝑠. 󵄨 󵄨

(60)

By Gronwall’s lemma, V = 0. Assume now that (𝐻5 ) is replaced by 𝑓1 ∈ 𝐿2 (𝑄𝑇 ); then 󸀠 } is bounded in 𝐿2 (0, 𝑇; 𝐻01 ). we only have to show that {V𝑚 󸀠 Indeed, multiplying the 𝑗th equation of (30) by 𝑐𝑚𝑗 (𝑡) and summing up with respect to 𝑗, afterwards, integrating with respect to the time variable from 0 to 𝑡, we get after some rearrangements 𝑡

󵄩 󵄩2 𝑋𝑚 (𝑡) = 𝑋𝑚 (0) + ∫ 𝜇󸀠 (𝑠) 󵄩󵄩󵄩V𝑚𝑥 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0

V (𝑥, 0) = ̃V0 (𝑥) , (0, 𝑇; 𝐻01 ) ,

(59)

󸀠

≤

(54)

+ ⟨𝑓 (V (𝑡) + 𝜑 (𝑡)) , 𝑤⟩ = ⟨𝑓1 (𝑡) , 𝑤⟩ ,

V∈𝐿

𝑡

𝑡

+ (𝜇 (𝑡) − 𝛼󸀠 (𝑡)) 𝑎 (V (𝑡) , 𝑤)

∞

𝑡

≥ −2𝛿 ∫ ‖V (𝑠)‖20 𝑑𝑠 ≥ −2𝛿 ∫ 𝜎1 (𝑠) 𝑑𝑠;

𝑡

1,1

𝜇, 𝛼 ∈ 𝑊

󸀠 + 2 ∫ ⟨𝑓1 (𝑠) , V𝑚 (𝑠)⟩ 𝑑𝑠

(0, 𝑇) .

0

(61)

𝑡

󸀠 − 2 ∫ ⟨𝑓 (V𝑚 (𝑠) + 𝜑 (𝑠)) , V𝑚 (𝑠)⟩ 𝑑𝑠,

Then,

0

󵄩2 󵄩 ‖V (𝑡)‖20 + 𝛼 (𝑡) 󵄩󵄩󵄩V𝑥 (𝑡)󵄩󵄩󵄩0 𝑡

where

󵄩 󵄩2 + ∫ (2𝜇 (𝑠) − 𝛼 (𝑠)) 󵄩󵄩󵄩V𝑥 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0 󸀠

(56)

𝑡

󵄩 󵄩2 󵄩 󵄩2 ̃ (𝑠) , V (𝑠)⟩ 𝑑𝑠. ≥ 󵄩󵄩󵄩̃V0 󵄩󵄩󵄩0 + 𝛼 (0) 󵄩󵄩󵄩̃V0𝑥 󵄩󵄩󵄩0 + 2 ∫ ⟨𝑓

𝑡 󵄩 󸀠 󵄩2 󵄩2 󵄩 󸀠 𝑋𝑚 (𝑡) = 2 ∫ (󵄩󵄩󵄩󵄩V𝑚 (𝑠)󵄩󵄩󵄩󵄩0 + 𝛼 (𝑠) 󵄩󵄩󵄩󵄩V𝑚𝑥 (𝑠)󵄩󵄩󵄩󵄩0 ) 𝑑𝑠 0

󵄩2 󵄩 + 𝜇 (𝑡) 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 .

0

Furthermore, if ̃V0 = 0, then the equality in (56) holds.

By the same estimates as above, we obtain

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 V1 and V2 be two weak solutions of (24). Then, V = V1 − V2 is a weak solution of (55) with the right-hand side function ̃ 𝑡) = −𝑓(V + 𝜑) + 𝑓(V + 𝜑) and ̃V = 0. Using replaced by 𝑓(𝑥, 1 2 0 Lemma 7, we have equality

󵄩 󵄩2 𝜇 (0) 𝑆; 𝑋𝑚 (0) = 𝜇 (0) 󵄩󵄩󵄩V0𝑚𝑥 󵄩󵄩󵄩0 ≤ 𝛼∗ 0 󵄨 󵄨 𝑡 𝑡 󵄨󵄨𝜇󸀠 (𝑠)󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩󵄩 󵄩󵄩2 󸀠 𝑆 (𝑠) 𝑑𝑠 ∫ 𝜇 (𝑠) 󵄩󵄩V𝑚𝑥 (𝑠)󵄩󵄩0 𝑑𝑠 ≤ ∫ 𝛼∗ 𝑚 0 0 ≤

𝑡

󵄩 󵄩2 𝜎1 (𝑡) = − ∫ (2𝜇 (𝑠) − 𝛼󸀠 (𝑠)) 󵄩󵄩󵄩V𝑥 (𝑠)󵄩󵄩󵄩0 𝑑𝑠 0

𝑡

𝑡

(57)

− 2 ∫ ⟨𝑓 (V1 + 𝜑) − 𝑓 (V2 + 𝜑) , V (𝑠)⟩ 𝑑𝑠,

󸀠 2 ∫ ⟨𝑓1 (𝑠) , V𝑚 (𝑠)⟩ 𝑑𝑠 0

1 𝑡󵄩 󸀠 󵄩2 󵄩 󵄩2 ≤ 2 󵄩󵄩󵄩󵄩𝑓1 󵄩󵄩󵄩󵄩𝐿2 (𝑄 ) + ∫ 󵄩󵄩󵄩󵄩V𝑚 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠 𝑇 2 0

0

where 󵄩2 󵄩 𝜎1 (𝑡) = ‖V (𝑡)‖20 + 𝛼 (𝑡) 󵄩󵄩󵄩V𝑥 (𝑡)󵄩󵄩󵄩0 .

𝐶𝑇 𝑇 󵄨󵄨 󸀠 󵄨󵄨 ∫ 󵄨󵄨𝜇 (𝑠)󵄨󵄨󵄨 𝑑𝑠; 𝛼∗ 0 󵄨

(58)

1 󵄩 󵄩2 ≤ 2 󵄩󵄩󵄩󵄩𝑓1 󵄩󵄩󵄩󵄩𝐿2 (𝑄 ) + 𝑋𝑚 (𝑡) ; 𝑇 4

(62)

8

Discrete Dynamics in Nature and Society 𝑡

(𝐻5󸀠 ) 𝑓1 ∈ 𝐿∞ (0, ∞; 𝐿2 ), and there exist the positive constants 𝐶1 , 𝛾1 and the function 𝑓1∞ ∈ 𝐿2 , such that ‖𝑓1 (𝑡) − 𝑓1∞ ‖0 ≤ 𝐶1 𝑒−𝛾1 𝑡 , ∀𝑡 ≥ 0.

󸀠 − 2 ∫ ⟨𝑓 (V𝑚 (𝑠) + 𝜑 (𝑠)) , V𝑚 (𝑠)⟩ 𝑑𝑠 0

≤ 2𝑇 (

1 𝑡󵄩 󸀠 𝑅𝛾+1 − 1 󵄩2 ) sup 𝑓2 (𝑧) + ∫ 󵄩󵄩󵄩󵄩V𝑚 (𝑠)󵄩󵄩󵄩󵄩0 𝑑𝑠 𝛾+1 2 0 |𝑧|≤𝐶

(𝐻6󸀠 ) 𝑓 ∈ 𝐶0 (R; R) and there exists a positive constant 𝛿, with 0 < 𝛿 < 2𝜇∞ /𝑅𝛾 (𝑅−1)2 , such that (𝑦−𝑧)(𝑓(𝑦)− 𝑓(𝑧)) ≥ −𝛿|𝑦 − 𝑧|2 , for all 𝑦, 𝑧 ∈ R.

𝑇

≤ 2𝑇 (

1 𝑅𝛾+1 − 1 ) sup 𝑓2 (𝑧) + 𝑋𝑚 (𝑡) . 𝛾+1 4 |𝑧|≤𝐶 𝑇

(63) This implies 𝑋𝑚 (𝑡) ≤

First, we consider the following stationary problem: −𝜇∞ (

𝑇 2 󵄨 󵄨 (𝜇 (0) 𝑆0 + 𝐶𝑇 ∫ 󵄨󵄨󵄨󵄨𝜇󸀠 (𝑠)󵄨󵄨󵄨󵄨 𝑑𝑠) 𝛼∗ 0

󵄩 󵄩2 + 4 󵄩󵄩󵄩󵄩𝑓1 󵄩󵄩󵄩󵄩𝐿2 (𝑄

1 < 𝑥 < 𝑅,

(1) 𝑅𝛾+1 − 1 ) sup 𝑓2 (𝑧) = 𝐶𝑇 . 𝛾+1 |𝑧|≤𝐶

(68)

𝑢 (1, 𝑡) = 𝑢 (𝑅, 𝑡) = 0.

(64)

𝑇)

+ 4𝑇 (

𝜕2 𝑢 𝛾 𝜕𝑢 + ) + 𝑓 (𝑢) = 𝑓1∞ (𝑥) , 𝜕𝑥2 𝑥 𝜕𝑥

The weak solution of problem (68) is obtained from the following variational problem. Find 𝑢∞ ∈ 𝐻01 such that

𝑇

󸀠 Then, the sequence {V𝑚 } is bounded in 𝐿2 (0, 𝑇; 𝐻01 ). Applying a similar argument used as above, the limit V of the sequence {V𝑚 } in suitable function spaces is a unique weak solution of (24) satisfying (28). Therefore, Theorem 6 is proved.

In this part, let 𝑇 > 0, (𝐻1 )–(𝐻6 ) hold. Then, there exists a unique solution 𝑢 = V + 𝜑 of problems (1)–(3) such that 𝑢−𝜑=V∈𝐿

(0, 𝑇; 𝐻01 ) ,

𝑡 (𝑢𝑡 − 𝜑𝑡 ) = 𝑡V𝑡 ∈ 𝐿2 (0, 𝑇; 𝐻01 ) .

(66)

(𝐻3󸀠 ) 𝛼 ∈ 𝑊1,1 (R+ ), 𝛼(𝑡) ≥ 𝛼∗ > 0, ∀𝑡 ≥ 0, 𝛼(0) ≤ 𝛼(𝑇). (𝐻4󸀠 ) 𝜇 ∈ 𝑊1,1 (R+ ), and there exist the positive constants 𝜇∗ , 𝜇∗ , 𝜇∞ , 𝐶𝜇 , and 𝛾𝜇 , such that 𝜇 (𝑡) ≥ 𝜇∗ > 0, ∀𝑡 ≥ 0,

󸀠

2𝜇 (𝑡) − 𝛼 (𝑡) ≥ 2𝜇∗ > 0, ∀𝑡 ≥ 0.

Proof. Consider the basis {𝑤𝑗 } for 𝐻01 as in Lemma 5. Put 𝑚

(70)

𝑗=1

(65)

̃ 0 (1) − 𝑔1 (0) = 𝑢 ̃ 0 (𝑅) − 𝑔𝑅 (0) = 0, (𝐻2󸀠 ) 𝑔1 , 𝑔𝑅 ∈ 𝑊1,1 (R+ ), 𝑢 and there exist the positive constants 𝐶1 , 𝐶𝑅 , 𝛾1 , and 𝛾𝑅 , such that

󵄨󵄨 󵄨 −𝛾 𝑡 󵄨󵄨𝜇 (𝑡) − 𝜇∞ 󵄨󵄨󵄨 ≤ 𝐶𝜇 𝑒 𝜇 , ∀𝑡 ≥ 0,

for all 𝑤 ∈ 𝐻01 , where 𝑎(⋅, ⋅) is the symmetric bilinear form on 𝐻01 × 𝐻01 defined by (21). We then have the following theorem.

𝑦𝑚 = ∑ 𝑑𝑚𝑗 𝑤𝑗 ,

We shall study asymptotic behavior of the solution 𝑢(𝑡) as 𝑡 → +∞. We make the following supplementary assumptions on the functions 𝑔1 (𝑡), 𝑔𝑅 (𝑡), 𝛼(𝑡), 𝜇(𝑡), and 𝑓1 (𝑥, 𝑡):

󵄨 󵄨 󸀠 󵄨 󵄨󵄨 −𝛾 𝑡 󵄨󵄨𝑔𝑖 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑔𝑖 (𝑡)󵄨󵄨󵄨󵄨 ≤ 𝐶𝑖 𝑒 𝑖 , ∀𝑡 ≥ 0, 𝑖 ∈ {1, 𝑅} .

(69)

Theorem 8. Let (𝐻4󸀠 )–(𝐻6󸀠 ) hold. Then, there exists a unique solution 𝑢∞ of the variational problem (69) such that 𝑢∞ ∈ 𝐻01 .

4. Asymptotic Behavior of the Solution as 𝑡 → +∞

∞

𝜇∞ 𝑎 (𝑢∞ , 𝑤) + ⟨𝑓 (𝑢∞ ) , 𝑤⟩ = ⟨𝑓1∞ , 𝑤⟩ ,

where 𝑑𝑚𝑗 satisfies the following nonlinear equation system: 𝜇∞ 𝑎 (𝑦𝑚 , 𝑤𝑗 ) + ⟨𝑓 (𝑦𝑚 ) , 𝑤𝑗 ⟩ = ⟨𝑓1∞ , 𝑤𝑗 ⟩ , 1 ≤ 𝑗 ≤ 𝑚.

By Brouwer’s lemma (see Lions [18], Lemma 4.3, p. 53), it follows from the hypotheses (𝐻4󸀠 )–(𝐻6󸀠 ) that systems (70) and (71) have a solution 𝑦𝑚 . Multiplying the 𝑗th equation of system (71) by 𝑑𝑚𝑗 , and then summing up with respect to 𝑗, we have 𝜇∞ 𝑎 (𝑦𝑚 , 𝑦𝑚 ) + ⟨𝑓 (𝑦𝑚 ) , 𝑦𝑚 ⟩ = ⟨𝑓1∞ , 𝑦𝑚 ⟩ . By using (𝐻6 ), we obtain 𝑅

⟨𝑓 (𝑦𝑚 ) , 𝑦𝑚 ⟩ = ∫ 𝑥𝛾 (𝑓 (𝑦𝑚 (𝑥)) − 𝑓 (0)) 𝑦𝑚 (𝑥) 𝑑𝑥 (67)

(71)

1

𝑅

+ ∫ 𝑥𝛾 𝑓 (0) 𝑦𝑚 (𝑥) 𝑑𝑥 1

(72)


9

𝑅

By means of (75) and Lemma 3, the sequence {𝑦𝑚 } has a subsequence still denoted by {𝑦𝑚 } such that

2 ≥ −𝛿 ∫ 𝑥𝛾 𝑦𝑚 (𝑥) 𝑑𝑥 1

𝑅

+ ∫ 𝑥𝛾 𝑓 (0) 𝑦𝑚 (𝑥) 𝑑𝑥 1

󵄩 󵄩2 󵄩 󵄩2 ≥ −𝛿 󵄩󵄩󵄩𝑦𝑚 󵄩󵄩󵄩0 − 𝜀1 󵄩󵄩󵄩𝑦𝑚 󵄩󵄩󵄩0 −

1 𝑅 𝛾 2 ∫ 𝑥 𝑓 (0) 𝑑𝑥 4𝜀1 1

1 𝑅𝛾+1 − 1 ( ) 𝑓2 (0) , 4𝜀1 𝛾+1

By using inequalities (20)(iii) and (73), we obtain from (72) that 󵄩 󵄩2 󵄩 󵄩2 𝜇∞ 󵄩󵄩󵄩𝑦𝑚𝑥 󵄩󵄩󵄩0 ≤ (𝛿 + 𝜀1 ) 󵄩󵄩󵄩𝑦𝑚 󵄩󵄩󵄩0 1 𝑅𝛾+1 − 1 ( ) 𝑓2 (0) 4𝜀1 𝛾+1

+

1 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩𝑓 󵄩󵄩 + 𝜀 󵄩󵄩𝑦 󵄩󵄩 4𝜀1 󵄩 1∞ 󵄩0 1 󵄩 𝑚 󵄩0

(74)

Equation (78) holds for every 𝑗 = 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.

(𝐻6󸀠󸀠 ) ∀𝑀 > 0, ∃𝑘𝑀 > 0 : |𝑓(𝑦) − 𝑓(𝑧)| ≤ 𝑘𝑀|𝑦 − 𝑧|, ∀𝑦, 𝑧 ∈ [−𝑀, 𝑀].

󵄩󵄩 󵄩 −𝛾𝑡 󵄩󵄩𝑢 (𝑡) − 𝑢∞ 󵄩󵄩󵄩1 ≤ 𝐶𝑒 , ∀𝑡 ≥ 0,

1 𝑅 −1 󵄩 󵄩2 [( ) 𝑓2 (0) + 󵄩󵄩󵄩𝑓1∞ 󵄩󵄩󵄩0 ] . 4𝜀1 𝛾+1

(79)

where 𝛾 > 0, 𝐶 > 0 are constants independent of 𝑡.

By 0 < 𝛿 < 2𝜇∞ /𝑅𝛾 (𝑅 − 1)2 , choose 𝜀1 > 0 such that (𝛿 + 2𝜀1 )(𝑅𝛾 /2)(𝑅 − 1)2 < 𝜇∞ . Hence, we deduce from (74) that

̃ 1 is a constant independent of 𝑚. and 𝐷

(78)

Then we have

𝛾+1

󵄩 󵄩2 ((𝑅𝛾+1 − 1) / (𝛾 + 1)) 𝑓2 (0) + 󵄩󵄩󵄩𝑓1∞ 󵄩󵄩󵄩0 󵄩󵄩 󵄩 󵄩󵄩𝑦𝑚𝑥 󵄩󵄩󵄩0 ≤ √ 4𝜀1 [𝜇∞ − (𝛿 + 2𝜀1 ) (𝑅𝛾 /2) (𝑅 − 1)2 ]

𝜇∞ 𝑎 (𝑢∞ , 𝑤𝑗 ) + ⟨𝑓 (𝑢∞ ) , 𝑤𝑗 ⟩ = ⟨𝑓1∞ , 𝑤𝑗 ⟩ .

(𝐻6󸀠󸀠󸀠 ) 0 < 𝛿 < (2/𝑅𝛾 (𝑅 − 1)2 )min{𝜇∞ , 𝜇∗ }.

1 𝑅𝛾+1 − 1 󵄩 󵄩2 [( ) 𝑓2 (0) + 󵄩󵄩󵄩𝑓1∞ 󵄩󵄩󵄩0 ] 4𝜀1 𝛾+1

̃1, =𝐷

(77)

And let 𝛿 > 0 in (𝐻6 ) satisfy the following condition, in addition,

𝑅𝛾 󵄩 󵄩2 = (𝛿 + 2𝜀1 ) (𝑅 − 1)2 󵄩󵄩󵄩𝑦𝑚𝑥 󵄩󵄩󵄩0 2 +

in 𝐶0 ([1, 𝑅]) strongly.

Theorem 9. Let (𝐻1 ), (𝐻2󸀠 )–(𝐻5󸀠 ), and (𝐻6 ) hold. Let 𝑓 satisfy the following condition, in addition,

󵄩 󵄩2 = (𝛿 + 2𝜀1 ) 󵄩󵄩󵄩𝑦𝑚 󵄩󵄩󵄩0 +

(76)

Now we consider asymptotic behavior of the solution 𝑢(𝑡) as 𝑡 → +∞. We then have the following theorem.

󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝑓1∞ 󵄩󵄩󵄩0 󵄩󵄩󵄩𝑦𝑚 󵄩󵄩󵄩0 󵄩 󵄩2 ≤ (𝛿 + 𝜀1 ) 󵄩󵄩󵄩𝑦𝑚 󵄩󵄩󵄩0 1 𝑅𝛾+1 − 1 ( ) 𝑓2 (0) 4𝜀1 𝛾+1


Passing to the limit in (71), we find without difficulty from (76) and (77) that 𝑢∞ satisfies the equation (73)

+

𝑦𝑚 󳨀→ 𝑢∞

𝑓 (𝑦𝑚 ) 󳨀→ 𝑓 (𝑢∞ )

∀𝜀1 > 0.

+

in 𝐻01 weakly,

On the other hand, by (76)2 and the continuity of 𝑓, we have

󵄩 󵄩2 = − (𝛿 + 𝜀1 ) 󵄩󵄩󵄩𝑦𝑚 󵄩󵄩󵄩0 −

𝑦𝑚 󳨀→ 𝑢∞

Proof. Put 𝑍𝑚 (𝑡) = V𝑚 (𝑡) − 𝑦𝑚 . Let us subtract (30)1 with (71) to obtain 󸀠 󸀠 ⟨𝑍𝑚 (𝑡) , 𝑤𝑗 ⟩ + 𝛼 (𝑡) 𝑎 (𝑍𝑚 (𝑡) , 𝑤𝑗 )

+ 𝜇 (𝑡) 𝑎 (𝑍𝑚 (𝑡) , 𝑤𝑗 ) + (𝜇 (𝑡) − 𝜇∞ ) 𝑎 (𝑦𝑚 , 𝑤𝑗 ) (75)

+ ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) − 𝑓 (𝑦𝑚 ) , 𝑤𝑗 ⟩ = ⟨𝑓1 (𝑡) − 𝑓1∞ , 𝑤𝑗 ⟩ , 1 ≤ 𝑗 ≤ 𝑚, 𝑍𝑚 (0) = V0𝑚 − 𝑦𝑚 .

(80)

10


By multiplying (80)1 by 𝑐𝑚𝑗 (𝑡) − 𝑑𝑚𝑗 and summing up in 𝑗, we obtain

Hence, 󵄩󵄩 󵄩2 󵄩󵄩𝑓 (𝑦𝑚 + 𝜑 (𝑡)) − 𝑓 (𝑦𝑚 )󵄩󵄩󵄩0

𝑑 󵄩󵄩 󵄩2 󵄩2 󵄩 [󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ] 𝑑𝑡 󵄩 󵄩2 + (2𝜇 (𝑡) − 𝛼󸀠 (𝑡)) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0

≤

+ 2 (𝜇 (𝑡) − 𝜇∞ ) 𝑎 (𝑦𝑚 , 𝑍𝑚 (𝑡))

+ 2 ⟨𝑓 (𝑦𝑚 + 𝜑 (𝑡)) − 𝑓 (𝑦𝑚 ) , 𝑍𝑚 (𝑡)⟩ = 2 ⟨𝑓1 (𝑡) − 𝑓1 (𝑡) , 𝑍𝑚 (𝑡)⟩ + 2 ⟨𝑓1 (𝑡) − 𝑓1∞ , 𝑍𝑚 (𝑡)⟩ . By the assumptions (𝐻2󸀠 )–(𝐻5󸀠 ), (𝐻6 ), (𝐻6󸀠󸀠 ), and (𝐻6󸀠󸀠󸀠 ) and using inequality (20)(iii), and with 𝜀1 > 0, we estimate without difficulty the following terms in (81) as follows: (i) Estimate (2𝜇(𝑡) − 𝛼󸀠 (𝑡))‖𝑍𝑚𝑥 (𝑡)‖20 , as 󵄩 󵄩2 󵄩2 󵄩 (2𝜇 (𝑡) − 𝛼 (𝑡)) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ≥ 2𝜇∗ 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ; 󸀠

(82)

(ii) Estimate 2(𝜇(𝑡) − 𝜇∞ )𝑎(𝑦𝑚 , 𝑍𝑚 (𝑡)), as

2 ⟨𝑓 (𝑦𝑚 + 𝜑 (𝑡)) − 𝑓 (𝑦𝑚 ) , 𝑍𝑚 (𝑡)⟩ ≤

1 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩𝑓 (𝑦𝑚 + 𝜑 (𝑡)) − 𝑓 (𝑦𝑚 )󵄩󵄩󵄩0 + 𝜀1 󵄩󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩0 𝜀1 󵄩

≤

1 1 2 Ψ2 (𝑡) (𝑅𝛾+1 − 1) 𝑘𝑀 1 𝜀1 𝛾 + 1 + 𝜀1

󵄩 󵄨󵄩 󵄩 󵄩 󵄨 ≤ 2 󵄨󵄨󵄨𝜇 (𝑡) − 𝜇∞ 󵄨󵄨󵄨 󵄩󵄩󵄩𝑦𝑚𝑥 󵄩󵄩󵄩0 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ̃ 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩 ≤ 2𝐶𝜇 𝑒−𝛾𝜇 𝑡 𝐶 󵄩0 󵄩

𝑓1 (𝑥, 𝑡) − 𝑓1 (𝑥, 𝑡) = −

+ (83)

2 ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) − 𝑓 (𝑦𝑚 + 𝜑 (𝑡)) , 𝑍𝑚 (𝑡)⟩ 𝛾

(84)

(iv) Estimate 2⟨𝑓(𝑦𝑚 + 𝜑(𝑡)) − 𝑓(𝑦𝑚 ), 𝑍𝑚 (𝑡)⟩. Note that, from the inequalities 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨𝜑 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑔1 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑔𝑅 (𝑡)󵄨󵄨󵄨 ≤ 𝐶1 + 𝐶𝑅 , 󵄩 󵄩 󵄩󵄩 󵄩󵄩 ̃1, 󵄩󵄩𝑦𝑚 󵄩󵄩𝐶0 (Ω) ≤ √𝑅 − 1 󵄩󵄩󵄩𝑦𝑚𝑥 󵄩󵄩󵄩 ≤ √𝑅 − 1𝐷 󵄩󵄩 󵄩 ̃ 1 + 𝐶1 + 𝐶𝑅 = 𝑀1 , 󵄩󵄩𝑦𝑚 + 𝜑󵄩󵄩󵄩𝐶0 (Ω) ≤ √𝑅 − 1𝐷

(85)

󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨𝑓1 (𝑥, 𝑡) − 𝑓1 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑔1󸀠 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑔𝑅󸀠 (𝑡)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝛾 (𝛼 (𝑡) + 𝜇 (𝑡)) + 𝑅−1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ⋅ [󵄨󵄨󵄨𝑔1 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑔1󸀠 (𝑡)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑔𝑅 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑔𝑅󸀠 (𝑡)󵄨󵄨󵄨󵄨] ≤ 𝐶1 𝑒−𝛾1 𝑡 𝛾 󵄩 󵄩 + 𝐶𝑅 𝑒−𝛾𝑅 𝑡 + (‖𝛼‖𝐿∞ (R+ ) + 󵄩󵄩󵄩𝜇󵄩󵄩󵄩𝐿∞ (R+ ) ) 𝑅−1

= [1 +

+ 𝐶𝑅 𝑒

.

(91)

𝛾 󵄩 󵄩 (‖𝛼‖𝐿∞ (R+ ) + 󵄩󵄩󵄩𝜇󵄩󵄩󵄩𝐿∞ (R+ ) )] 𝑅−1

⋅ [𝐶1 𝑒−𝛾1 𝑡 + 𝐶𝑅 𝑒−𝛾𝑅 𝑡 ]

(86)

𝛾 󵄩 󵄩 (‖𝛼‖𝐿∞ (R+ ) + 󵄩󵄩󵄩𝜇󵄩󵄩󵄩𝐿∞ (R+ ) )] Ψ (𝑡) 𝑅−1

̃ (𝛼, 𝜇) Ψ (𝑡) . ≡𝐶 It follows that

where Ψ (𝑡) = 𝐶1 𝑒

(90)

+ 𝛼 (𝑡) (𝑔𝑅󸀠 (𝑡) − 𝑔1󸀠 (𝑡))] .

= [1 +

−𝛾𝑅 𝑡

𝛾 [𝜇 (𝑡) (𝑔𝑅 (𝑡) − 𝑔1 (𝑡)) (𝑅 − 1) 𝑥

⋅ [𝐶1 𝑒−𝛾1 𝑡 + 𝐶𝑅 𝑒−𝛾𝑅 𝑡 ]

and (𝐻6󸀠󸀠 ), we deduce that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨𝑓 (𝑦𝑚 + 𝜑 (𝑡)) − 𝑓 (𝑦𝑚 )󵄨󵄨󵄨 ≤ 𝑘𝑀1 󵄨󵄨󵄨𝜑 (𝑥, 𝑡)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 𝑘𝑀1 (󵄨󵄨󵄨𝑔1 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑔𝑅 (𝑡)󵄨󵄨󵄨) = 𝑘𝑀1 Ψ (𝑡) ,

1 [(𝑥 − 1) 𝑔𝑅󸀠 (𝑡) 𝑅−1

Hence,

(iii) Estimate 2⟨𝑓(V𝑚 (𝑡) + 𝜑(𝑡)) − 𝑓(𝑦𝑚 + 𝜑(𝑡)), 𝑍𝑚 (𝑡)⟩, as

𝑅 󵄩 󵄩2 󵄩2 󵄩 ≥ −2𝛿 󵄩󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩0 ≥ −2𝛿 (𝑅 − 1)2 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 . 2

𝑅𝛾 󵄩2 󵄩 (𝑅 − 1)2 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 . 2

+ (𝑅 − 𝑥) 𝑔1󸀠 (𝑡)]

1 ̃2 2 −2𝛾𝜇 𝑡 󵄩2 󵄩 𝐶 𝐶𝜇 𝑒 + 𝜀1 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 . 𝜀1

(89)

(v) Estimate ⟨𝑓1 (𝑡) − 𝑓1 (𝑡), 𝑍𝑚 (𝑡)⟩. We have

2 (𝜇 (𝑡) − 𝜇∞ ) 𝑎 (𝑦𝑚 , 𝑍𝑚 (𝑡))

−𝛾1 𝑡

(88)

Thus,

+ 2 ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) − 𝑓 (𝑦𝑚 + 𝜑 (𝑡)) , 𝑍𝑚 (𝑡)⟩ (81)

≤

1 2 Ψ2 (𝑡) . (𝑅𝛾+1 − 1) 𝑘𝑀 1 𝛾+1

(87)

1 󵄩󵄩 󵄩2 𝛾+1 2 ̃2 󵄩󵄩𝑓1 (𝑡) − 𝑓1 (𝑡)󵄩󵄩󵄩 ≤ 󵄩 󵄩0 𝛾 + 1 (𝑅 − 1) 𝐶 (𝛼, 𝜇) Ψ (𝑡) . (92)


11

Thus,

≥ 2 ⟨𝑓1 (𝑡) − 𝑓1 (𝑡) , 𝑍𝑚 (𝑡)⟩ 1 ≤ 𝜀1

+

󵄩󵄩 󵄩2 󵄩󵄩𝑓1 (𝑡) − 𝑓1 (𝑡)󵄩󵄩󵄩 + 𝜀1 󵄩󵄩󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩󵄩20 󵄩 󵄩0

1 1 ̃2 (𝛼, 𝜇) Ψ2 (𝑡) ≤ (𝑅𝛾+1 − 1) 𝐶 𝜀1 𝛾 + 1 + 𝜀1

(97) where 𝛽1 = (1/2)min{1/‖𝛼‖𝐿∞ (R+ ) , 2/𝑅𝛾 (𝑅 − 1)2 }, it follows from (96) and (97) that 𝑑 󵄩󵄩 󵄩2 󵄩2 󵄩 [󵄩𝑍 (𝑡)󵄩󵄩 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ] 𝑑𝑡 󵄩 𝑚 󵄩0 󵄩2 󵄩2 󵄩 󵄩 ̃ (𝑡) + 2̃𝛾𝛽1 (󵄩󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ) ≤ 𝜓

(vi) Estimate ⟨𝑓1 (𝑡) − 𝑓1∞ , 𝑍𝑚 (𝑡)⟩, as 2 ⟨𝑓1 (𝑡) − 𝑓1∞ , 𝑍𝑚 (𝑡)⟩ 1 󵄩󵄩 󵄩2 󵄩2 󵄩 󵄩𝑓 (𝑡) − 𝑓1∞ 󵄩󵄩󵄩0 + 𝜀1 󵄩󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩0 𝜀1 󵄩 1

≤

𝑅𝛾 1 2 −2𝛾1 𝑡 󵄩2 󵄩 𝐶1 𝑒 + 𝜀1 (𝑅 − 1)2 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 . 𝜀1 2

Choose 𝛾 > 0 such that 𝛾 < min{𝛾0 , 2̃𝛾𝛽1 }, and then we have from (98) that 𝑑 󵄩󵄩 󵄩2 󵄩2 󵄩 [󵄩𝑍 (𝑡)󵄩󵄩 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ] 𝑑𝑡 󵄩 𝑚 󵄩0 󵄩2 󵄩2 󵄩 󵄩 + 2𝛾 (󵄩󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ) ≤ 𝐶0 𝑒−2𝛾0 𝑡 .

𝑑 󵄩󵄩 󵄩2 󵄩2 󵄩 [󵄩𝑍 (𝑡)󵄩󵄩 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ] 𝑑𝑡 󵄩 𝑚 󵄩0 𝑅𝛾 󵄩2 󵄩 (𝑅 − 1)2 ] 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 2

1 ̃2 2 −2𝛾𝜇 𝑡 ≤ (𝐶 𝐶𝜇 𝑒 + 𝐶12 𝑒−2𝛾1 𝑡 ) 𝜀1 +

󵄩2 󵄩2 󵄩 󵄩2 󵄩 󵄩󵄩 󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼∗ 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ≤ 󵄩󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼 (𝑡) 󵄩2 󵄩 ⋅ 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0

(95)

󵄩2 󵄩2 󵄩 󵄩 ≤ (󵄩󵄩󵄩𝑍𝑚 (0)󵄩󵄩󵄩0 + 𝛼 (0) 󵄩󵄩󵄩𝑍𝑚𝑥 (0)󵄩󵄩󵄩0 +

By 0 < 𝛿 < (2/𝑅𝛾 (𝑅−1)2 ) min{𝜇∞ , 𝜇∗ } ≤ 2𝜇∗ /𝑅𝛾 (𝑅−1)2 , choose 𝜀1 > 0 such that 2̃𝛾 = 2𝜇∗ − 𝜀1 − (2𝛿 + 3𝜀1 )(𝑅𝛾 /2)(𝑅 − 1)2 > 0. ̃ (𝑡) ≤ 𝐶0 𝑒−2𝛾0 𝑡 Put 𝛾0 = min{𝛾1 , 𝛾1 , 𝛾𝑅 , 𝛾𝜇 }, and we have 𝜓 for all 𝑡 ≥ 0, as

̃2 = √ 𝐷

𝐶0 ) 2 (𝛾0 − 𝛾)

(100)

⋅ 𝑒−2𝛾𝑡 . Letting 𝑚 → +∞ in (100), we obtain 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩V (𝑡) − 𝑢∞ 󵄩󵄩󵄩0 + 𝛼∗ 󵄩󵄩󵄩V𝑥 (𝑡) − 𝑢∞𝑥 󵄩󵄩󵄩0 󵄩2 󵄩 󵄩2 󵄩 ≤ lim inf (󵄩󵄩󵄩V𝑚 (𝑡) − 𝑦𝑚 󵄩󵄩󵄩0 + 𝛼∗ 󵄩󵄩󵄩V𝑚𝑥 (𝑡) − 𝑦𝑚𝑥 󵄩󵄩󵄩0 ) 𝑚→+∞

(96)

󵄩2 󵄩2 󵄩 󵄩 ≤ (󵄩󵄩󵄩̃V0 − 𝑢∞ 󵄩󵄩󵄩0 + 𝛼 (0) 󵄩󵄩󵄩̃V0𝑥 − 𝑢∞𝑥 󵄩󵄩󵄩0

By 󵄩2 1 󵄩 󵄩2 1 󵄩 󵄩2 󵄩󵄩 󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 = 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 + 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 2 2 1 1 󵄩2 󵄩 ≥ 𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 2 𝛼 (𝑡) 2 1 󵄩2 󵄩󵄩 + 󵄩𝑍 (𝑡)󵄩󵄩 2 𝑅𝛾 (𝑅 − 1)2 󵄩 𝑚 󵄩0

(99)

Hence, we obtain from (99) that

1 𝑅𝛾+1 − 1 2 ̃2 (𝛼, 𝜇)) Ψ2 (𝑡) ≡ 𝜓 ̃ (𝑡) . (𝑘𝑀1 + 𝐶 𝜀1 𝛾 + 1

𝑑 󵄩󵄩 󵄩2 󵄩2 󵄩2 󵄩 󵄩 [󵄩𝑍 (𝑡)󵄩󵄩 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ] + 2̃𝛾 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 𝑑𝑡 󵄩 𝑚 󵄩0 ̃ (𝑡) ≤ 𝐶0 𝑒−2𝛾0 𝑡 . ≤𝜓

(98)

≤ 𝐶0 𝑒−2𝛾0 𝑡 .

(94)

It follows from (81)–(84), (89), (93), and (94) that

+ [2𝜇∗ − 𝜀1 − (2𝛿 + 3𝜀1 )

2 1 󵄩2 󵄩󵄩 󵄩𝑍 (𝑡)󵄩󵄩 2 𝑅𝛾 (𝑅 − 1)2 󵄩 𝑚 󵄩0

󵄩2 󵄩 󵄩2 󵄩 ≥ 𝛽1 (𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 + 󵄩󵄩󵄩𝑍𝑚 (𝑡)󵄩󵄩󵄩0 ) ,

(93)

𝑅𝛾 󵄩2 󵄩 (𝑅 − 1)2 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 . 2

≤

1 1 󵄩2 󵄩 𝛼 (𝑡) 󵄩󵄩󵄩𝑍𝑚𝑥 (𝑡)󵄩󵄩󵄩0 2 ‖𝛼‖𝐿∞ (R+ )

+

𝐶0 ) 𝑒−2𝛾𝑡 , 2 (𝛾0 − 𝛾)

(101)

∀𝑡 ≥ 0,

or 󵄩 󵄩󵄩 ̃ 2 𝑒−𝛾𝑡 , ∀𝑡 ≥ 0, 󵄩󵄩V (𝑡) − 𝑢∞ 󵄩󵄩󵄩1 ≤ 𝐷

(102)

where

𝐶0 1 󵄩2 󵄩2 󵄩 󵄩 (󵄩󵄩󵄩̃V0 − 𝑢∞ 󵄩󵄩󵄩0 + 𝛼 (0) 󵄩󵄩󵄩̃V0𝑥 − 𝑢∞𝑥 󵄩󵄩󵄩0 + ). min (1, 𝛼∗ ) 2 (𝛾0 − 𝛾)

(103)

12

Discrete Dynamics in Nature and Society where 𝑝 > 0, 𝜁𝑘 , and 𝑘 ∈ {1, 𝑅} are constants. It is obvious that (𝐻2 ) holds, because

Note that 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 −𝛾 𝑡 −𝛾 𝑡 󵄨󵄨𝜑 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑔𝑅 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑔1 (𝑡)󵄨󵄨󵄨 ≤ 𝐶1 𝑒 1 + 𝐶𝑅 𝑒 𝑅 = Ψ (𝑡) ; 󵄨 󵄨 1 󵄨 󵄨󵄨𝑔𝑅 (𝑡) − 𝑔1 (𝑡)󵄨󵄨󵄨 󵄨󵄨 ≤ Ψ (𝑡) . 󵄨󵄨𝜑𝑥 (𝑥, 𝑡)󵄨󵄨󵄨 = 󵄨 𝑅−1 𝑅−1

(104)

󵄩2 󵄩2 󵄩 󵄩󵄩 󵄩2 󵄩 󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩1 = 󵄩󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩0 + 󵄩󵄩󵄩𝜑𝑥 (𝑡)󵄩󵄩󵄩0 1 1 ) Ψ2 (𝑡) (𝑅𝛾+1 − 1) (1 + 𝛾+1 (𝑅 − 1)2

(105)

2

󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑢 (𝑡) − 𝑢∞ 󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩V (𝑡) − 𝑢∞ 󵄩󵄩󵄩1 + 󵄩󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩1 ∀𝑡 ≥ 0.

(106)

V (𝑥, 0) = 𝜂V (𝑥, 𝑇) , where 𝑓1 (𝑥, 𝑡) is defined by (25)1 . The weak formulation of problem (112) can be given in the following manner: Find V ∈ 𝐿∞ (0, 𝑇; 𝐻01 ) with V𝑡 ∈ 𝐿2 (0, 𝑇; 𝐻01 ), such that V satisfies the following variational equation:

5. The Existence and the Uniqueness of a (𝜂,𝑇)-Periodic Weak Solution In this section, we shall consider problems (1), (2), and (4) with 𝑅 > 1, 0 < |𝜂| ≤ 1 as given constants and 𝜇, 𝛼, 𝑓, 𝑓1 , 𝑔1 , and 𝑔𝑅 as given functions satisfying the following assumptions: (𝐻2 ) 𝑔1 , 𝑔𝑅 ∈ 𝑊1,1 (0, 𝑇), and 𝑔1 and 𝑔𝑅 are (𝜂, 𝑇)-periodic; that is,

𝑇

𝑇

∫ ⟨V󸀠 (𝑡) , 𝑤 (𝑡)⟩ 𝑑𝑡 + ∫ 𝛼 (𝑡) 𝑎 (V󸀠 (𝑡) , 𝑤 (𝑡)) 𝑑𝑡 0

0

𝑇

+ ∫ 𝜇 (𝑡) 𝑎 (V (𝑡) , 𝑤 (𝑡)) 𝑑𝑡 0

𝑇

+ ∫ ⟨𝑓 (V (𝑡) + 𝜑 (𝑡)) , 𝑤 (𝑡)⟩ 𝑑𝑡 𝑇

= ∫ ⟨𝑓1 (𝑡) , 𝑤 (𝑡)⟩ 𝑑𝑡, 0

(𝐻3 ) 𝛼 ∈ 𝑊1,1 (0, 𝑇), 𝛼(𝑡) ≥ 𝛼∗ > 0, ∀𝑡 ∈ [0, 𝑇], 0 < 𝛼(0) ≤ 𝛼(𝑇); (𝐻4 ) 𝜇 ∈ 𝑊1,1 (0, 𝑇), 𝜇(𝑡) ≥ 𝜇∗ > 0, ∀𝑡 ∈ [0, 𝑇], 2𝜇(𝑡) − 𝛼󸀠 (𝑡) ≥ 2𝜇∗ > 0, a.e., 𝑡 ∈ [0, 𝑇]; (𝐻5 ) 𝑓1 , 𝑓1󸀠 ∈ 𝐿2 (𝑄𝑇 ), 𝑓1 is (𝜂, 𝑇)-periodic, 𝑓1 (0) = 𝜂𝑓1 (𝑇); (𝐻6 ) 𝑓 ∈ 𝐶0 (R; R) and there exists a positive constant 𝛿, with 2𝜇∗

𝑅𝛾 (𝑅 − 1)2

,

∀𝑤 ∈ 𝐿2 (0, 𝑇; 𝐻01 ) ,

V (0) = 𝜂V (𝑇) , where 𝑎(⋅, ⋅) is the symmetric bilinear form on 𝐻01 × 𝐻01 defined by (21). Then, we have the following theorem. Theorem 11. Let 𝑇 > 0 and (𝐻2 )–(𝐻6 ) hold. Then, problem (112) has a unique weak solution V such that V ∈ 𝐿∞ (0, 𝑇; 𝐻01 ) ,

󵄨 󵄨2 such that (𝑦 − 𝑧) (𝑓 (𝑦) − 𝑓 (𝑧)) ≥ −𝛿 󵄨󵄨󵄨𝑦 − 𝑧󵄨󵄨󵄨 ,

(108)

∀𝑦, 𝑧 ∈ R.

V𝑡 ∈ 𝐿2 (0, 𝑇; 𝐻01 ) .

(114)

Proof. The proof consists of several steps.

Remark 10. An example of the functions 𝑔1 , 𝑔𝑅 satisfying (𝐻2 ) is 𝑔𝑘 (𝑡) = 𝜁𝑘 𝑒𝑝𝑡 ,

(113)

0

(107)

𝑔𝑅 (0) = 𝜂𝑔𝑅 (𝑇) .

0 0 such that 𝛾̃ = 𝜇∗ − (𝛿 + 𝜀1 )(𝑅𝛾 /2)(𝑅 − 1)2 > 0. It is similar to (97); we get 󵄩󵄩 󵄩2 󵄩2 󵄩 󵄩2 󵄩 󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ≥ 𝛽1 (𝛼 (𝑡) 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 + 󵄩󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩0 ) , where 𝛽1 = (1/2) min{1/‖𝛼‖𝐿∞ (0,𝑇) , 2/𝑅𝛾 (𝑅 − 1)2 }.

(124)

Let us define

Hence, it follows from (116) and (117) that

𝑅𝛾 󵄩2 󵄩 (𝑅 − 1)2 ] 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 2

(123) ∀𝑚.

𝑅𝛾 󵄩󵄩 󵄩2 󵄩󵄩𝑓1 (𝑡)󵄩󵄩󵄩 + 𝜀1 (𝑅 − 1)2 󵄩󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩󵄩20 . 󵄩 󵄩0 2

+ 2 [𝜇∗ − (𝛿 + 𝜀1 )

(122)

󵄩2 󵄩2 󵄩󵄩 󵄩 2 󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼 (𝑡) 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ≤ 𝜌 ,

+ 2 ⟨𝑓 (𝜑 (𝑡)) , V𝑚 (𝑡)⟩

2 ⟨𝑓1 (𝑡) , V𝑚 (𝑡)⟩ ≤

𝑡 = 0.

Therefore, if we choose V0𝑚 such that ‖V0𝑚 ‖20 +𝛼(0)‖V0𝑚𝑥 ‖20 ≤ 𝜌 , we obtain from (121) that

= 2 ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) − 𝑓 (𝜑 (𝑡)) , V𝑚 (𝑡)⟩

1 󵄩 󵄩2 − 󵄩󵄩󵄩𝑓 (𝜑 (𝑡))󵄩󵄩󵄩0 ; 𝜀1

0 < 𝑡 ≤ 𝑇,

2

2 ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) , V𝑚 (𝑡)⟩

𝑅𝛾 󵄩2 󵄩 (𝑅 − 1)2 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 2

(121)

2

⋅ 𝑒−2𝛽1 𝛾̃𝑡 ,

+ 2 ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) , V𝑚 (𝑡)⟩

≥ − (2𝛿 + 𝜀1 )

(120)

(119)

(125)

We prove that F𝑚 is a contraction. Let V0𝑚 , V0𝑚 ∈ 𝐵𝑚 (0, 𝜌) and let 𝑦𝑚 (𝑡) = V𝑚 (𝑡) − V𝑚 (𝑡), where V𝑚 (𝑡) and V𝑚 (𝑡) are solutions of system (30)1 on [0, 𝑇] satisfying the initial conditions V𝑚 (0) = V0𝑚 and V𝑚 (0) = V0𝑚 , respectively. Then, 𝑦𝑚 (𝑡) satisfies the following differential equation system: 󸀠 󸀠 ⟨𝑦𝑚 (𝑡) , 𝑤𝑗 ⟩ + 𝛼 (𝑡) 𝑎 (𝑦𝑚 (𝑡) , 𝑤𝑗 )

+ 𝜇 (𝑡) 𝑎 (𝑦𝑚 (𝑡) , 𝑤𝑗 ) + ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) − 𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) , 𝑤𝑗 ⟩ = 0,

(126)

14


where 1 ≤ 𝑗 ≤ 𝑚, with initial condition 𝑦𝑚 (0) = V0𝑚 − V0𝑚 .

(127)

We estimate without difficulty the following terms in (132): −∫

By using the same arguments as before, we can show that 𝑑 󵄩󵄩 󵄩2 󵄩2 󵄩 [󵄩𝑦 (𝑡)󵄩󵄩 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑦𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ] 𝑑𝑡 󵄩 𝑚 󵄩0 󵄩2 󵄩2 󵄩 󵄩 + 2𝛽2 [󵄩󵄩󵄩𝑦𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑦𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ] ≤ 0,

0

(128)

Note that 󵄩󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩𝐶0 ([1,𝑅]) ≤ 󵄨󵄨󵄨𝑔1 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑔𝑅 (𝑡)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ sup (󵄨󵄨󵄨𝑔1 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑔𝑅 (𝑡)󵄨󵄨󵄨) , 󵄩󵄩 󵄩 󵄩󵄩V𝑚 (𝑡) + 𝜑 (𝑡)󵄩󵄩󵄩𝐶0 ([1,𝑅]) 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩𝐶0 ([1,𝑅]) + 󵄩󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩𝐶0 ([1,𝑅]) 𝜌 󵄨 󵄨 󵄨 󵄨 ≤ √𝑅 − 1 + sup (󵄨󵄨𝑔1 (𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑔𝑅 (𝑡)󵄨󵄨󵄨) √𝛼∗ 0≤𝑡≤𝑇 󵄨

(129)

󵄩2 󵄩2 󵄩2 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑦𝑚 (𝑇)󵄩󵄩󵄩𝑉𝑚 = 󵄩󵄩󵄩𝑦𝑚 (𝑇)󵄩󵄩󵄩0 + 𝛼 (0) 󵄩󵄩󵄩𝑦𝑚𝑥 (𝑇)󵄩󵄩󵄩0

Hence, 𝑇

󸀠 2 ∫ ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) , V𝑚 (𝑡)⟩ 𝑑𝑡

(130)

0

𝑇

󵄩2 󵄩 ≤ 2 ∫ 󵄩󵄩󵄩𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡))󵄩󵄩󵄩0 𝑑𝑡 0

or 󵄩 󵄩 󵄨 󵄨 ≤ 󵄨󵄨󵄨𝜂󵄨󵄨󵄨 𝑒−𝛽2 𝑇 󵄩󵄩󵄩V0𝑚 − V0𝑚 󵄩󵄩󵄩𝑉𝑚 ;

1 𝑇󵄩 󸀠 󵄩2 + ∫ 󵄩󵄩󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩󵄩0 𝑑𝑡 2 0 (131) ≤ 𝑀2 (𝑇) +

that is, F𝑚 is a contraction. Therefore, there exists a unique function V0𝑚 ∈ 𝐵𝑚 (𝜌) such that the solution of the initial value problem (30) is a solution of system (30)1 , (115). This solution satisfies inequality (124) a.e., in [0, 𝑇]. On the other hand, multiplying the 𝑗th equation of (30)1 󸀠 by 𝑐𝑚𝑗 (𝑡) and summing up with respect to 𝑗, afterwards, integrating with respect to the time variable from 0 to 𝑇, we get after some rearrangements 𝑇

󵄩 󸀠 󵄩2 󵄩2 󵄩 󸀠 2 ∫ (󵄩󵄩󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩󵄩0 + 𝛼 (𝑡) 󵄩󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩󵄩0 ) 𝑑𝑡 0 +∫

𝑇

0

𝑇

0

𝑇

󵄩 󵄩2 = ∫ 𝜇 (𝑡) 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 𝑑𝑡 0 󸀠

+ 2∫

𝑇

0

󸀠 ⟨𝑓1 (𝑡) , V𝑚

(𝑡)⟩ 𝑑𝑡.

(136)

1 𝑇 󵄩󵄩 󸀠 󵄩2 ∫ 󵄩󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩󵄩0 𝑑𝑡, 2 0

where 𝑀2 (𝑇) = 2𝑇 (

𝑅𝛾+1 − 1 ) sup 𝑓2 (𝑧) . 𝛾+1 |𝑧|≤𝑀1 (𝑇)

(137)

Moreover, 𝑇

󸀠 2 ∫ ⟨𝑓1 (𝑡) , V𝑚 (𝑡)⟩ 𝑑𝑡 0

𝑇 1 𝑇󵄩 󸀠 󵄩 󵄩2 󵄩2 ≤ 2 ∫ 󵄩󵄩󵄩󵄩𝑓1 (𝑡)󵄩󵄩󵄩󵄩0 𝑑𝑡 + ∫ 󵄩󵄩󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩󵄩0 𝑑𝑡; 2 0 0

(138)

𝑇 𝜌2 𝑇 󵄨󵄨 󸀠 󵄨󵄨 󵄩 󵄩2 ∫ 𝜇󸀠 (𝑡) 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 𝑑𝑡 ≤ ∫ 󵄨󵄨𝜇 (𝑡)󵄨󵄨󵄨 𝑑𝑡. 𝛼∗ 0 󵄨 0

𝑑 󵄩2 󵄩 (𝜇 (𝑡) 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ) 𝑑𝑡 𝑑𝑡

󸀠 + 2 ∫ ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) , V𝑚 (𝑡)⟩ 𝑑𝑡

(135)

= 𝑀1 (𝑇) .

󵄩 󵄩2 = 𝑒−2𝛽2 𝑇 󵄩󵄩󵄩V0𝑚 − V0𝑚 󵄩󵄩󵄩𝑉𝑚 , 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩F𝑚 (V0𝑚 ) − F𝑚 (V0𝑚 )󵄩󵄩󵄩𝑉𝑚 = 󵄩󵄩󵄩𝜂𝑦𝑚 (𝑇)󵄩󵄩󵄩𝑉𝑚

(134)

0≤𝑡≤𝑇

By 0 < 𝛼(0) ≤ 𝛼(𝑇), it follows that

󵄩2 󵄩 ≤ 𝑒−2𝛽2 𝑇 󵄩󵄩󵄩𝑦𝑚 (0)󵄩󵄩󵄩𝑉𝑚

(133)

𝜌2 󵄩2 󵄩 ≤ 𝜇 (0) 󵄩󵄩󵄩V𝑚𝑥 (0)󵄩󵄩󵄩0 ≤ 𝜇 (0) . 𝛼∗

󵄩2 󵄩 = 𝑒−2𝛽2 𝑡 󵄩󵄩󵄩𝑦𝑚 (0)󵄩󵄩󵄩𝑉𝑚 , ∀𝑡 ∈ [0, 𝑇] .

󵄩2 󵄩2 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑦𝑚 (𝑇)󵄩󵄩󵄩0 + 𝛼 (𝑇) 󵄩󵄩󵄩𝑦𝑚𝑥 (𝑇)󵄩󵄩󵄩0

𝑑 󵄩2 󵄩 (𝜇 (𝑡) 󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩0 ) 𝑑𝑡 𝑑𝑡

󵄩2 󵄩2 󵄩 󵄩 = −𝜇 (𝑇) 󵄩󵄩󵄩V𝑚𝑥 (𝑇)󵄩󵄩󵄩0 + 𝜇 (0) 󵄩󵄩󵄩V𝑚𝑥 (0)󵄩󵄩󵄩0

where 𝛽2 = 𝛽1 (𝜇∗ − 𝛿(𝑅𝛾 /2)(𝑅 − 1)2 ) > 0, 𝛽1 = (1/2)min{1/ ‖𝛼‖𝐿∞ (0,𝑇) , 2/𝑅𝛾 (𝑅 − 1)2 }. Integrating inequality (128), we obtain 󵄩2 󵄩2 󵄩󵄩 󵄩 󵄩󵄩𝑦𝑚 (𝑡)󵄩󵄩󵄩0 + 𝛼 (𝑡) 󵄩󵄩󵄩𝑦𝑚𝑥 (𝑡)󵄩󵄩󵄩0 󵄩2 󵄩2 󵄩 󵄩 ≤ 𝑒−2𝛽2 𝑡 (󵄩󵄩󵄩𝑦𝑚 (0)󵄩󵄩󵄩0 + 𝛼 (0) 󵄩󵄩󵄩𝑦𝑚𝑥 (0)󵄩󵄩󵄩0 )

𝑇

It follows from (132), (133), (136), and (138) that (132)

𝑇 𝑇 󵄩 󸀠 󵄩 󸀠 󵄩2 󵄩2 ∫ 󵄩󵄩󵄩󵄩V𝑚 (𝑡)󵄩󵄩󵄩󵄩0 𝑑𝑡 + 2𝛼∗ ∫ 󵄩󵄩󵄩󵄩V𝑚𝑥 (𝑡)󵄩󵄩󵄩󵄩0 𝑑𝑡 0 0

≤ 𝜇 (0) +

𝑇 𝜌2 󵄩 󵄩2 + 𝑀2 (𝑇) + 2 ∫ 󵄩󵄩󵄩󵄩𝑓1 (𝑡)󵄩󵄩󵄩󵄩0 𝑑𝑡 𝛼∗ 0

𝜌2 𝑇 󵄨󵄨 󸀠 󵄨󵄨 ∫ 󵄨󵄨𝜇 (𝑡)󵄨󵄨󵄨 𝑑𝑡 ≤ 𝐶𝑇 , 𝛼∗ 0 󵄨

(139)


15

for all 𝑚 ∈ N, for all 𝑡 ∈ [0, 𝑇], where 𝐶𝑇 always indicates a bound depending on 𝑇. Step 3 (the limiting process). By (123) and (139), we deduce that there exists a subsequence of {V𝑚 }, still denoted by {V𝑚 } such that V𝑚 󳨀→ V

∞

in 𝐿

󸀠 󳨀→ V󸀠 V𝑚

(0, 𝑇; 𝐻01 )

V𝑚 (𝑥, 𝑡) 󳨀→ V (𝑥, 𝑡)

weakly ,

in 𝐿2 (0, 𝑇; 𝐻01 ) weakly.

a.e., (𝑥, 𝑡) in 𝑄𝑇 ,

(150)

Because 𝑓 is continuous, then

∗

𝑓 (V𝑚 (𝑥, 𝑡) + 𝜑 (𝑥, 𝑡)) 󳨀→

(140)

𝑓 (V (𝑥, 𝑡) + 𝜑 (𝑥, 𝑡))

From (115), we obtain

a.e., (𝑥, 𝑡) in 𝑄𝑇 .

(151)

On the other hand,

V (0) = 𝜂V (𝑇) .

(141)

Indeed, we prove (141) as follows. By ‖V0𝑚𝑥 ‖ ≤ (1/√𝛼∗ )‖V0𝑚 ‖𝑉𝑚 ≤ 𝜌/√𝛼∗ , and by the imbedding 𝐻01 󳨅→ 𝐶0 (Ω) being compact, there exists a subsequence of {V0𝑚 }, still denoted by {V0𝑚 } such that V0𝑚 󳨀→ ̃V0

in 𝐻01 weakly,

V0𝑚 󳨀→ ̃V0


By V𝑚 (𝑡) = (142) that

By the Riesz-Fischer theorem, we can extract from {V𝑚 } a subsequence still denoted by {V𝑚 }, such that

𝑡 󸀠 V𝑚 (0) + ∫0 V𝑚 (𝑠)𝑑𝑠,

(142)

𝑡

(143)

0

󵄨󵄨 󵄨 󵄨󵄨𝑓 (𝑧)󵄨󵄨󵄨 ,

sup

|𝑧|≤𝑀1 (𝑇)

(152) a.e., (𝑥, 𝑡) in 𝑄𝑇 ,

where 𝑀1 (𝑇) is the constant defined by (135). Using the dominated convergence theorem, (151) and (152) yield 𝑓 (V𝑚 + 𝜑) 󳨀→ 𝑓 (V + 𝜑)

we deduce from (140) and

V (𝑡) = ̃V0 + ∫ V󸀠 (𝑠) 𝑑𝑠.

󵄨󵄨 󵄨 󵄨󵄨𝑓 (V𝑚 (𝑥, 𝑡) + 𝜑 (𝑥, 𝑡))󵄨󵄨󵄨 ≤


(153)

Denote by {𝜁𝑖 , 𝑖 = 1, 2, . . .} the orthonormal base in the real Hilbert space 𝐿2 (0, 𝑇). The set {𝜁𝑖 𝑤𝑗 , 𝑖, 𝑗 = 1, 2, . . .} forms an orthonormal base in 𝐿2 (0, 𝑇; 𝐻01 ). From (30)1 we have 𝑇

󸀠 ∫ ⟨V𝑚 (𝑡) , 𝑤𝑗 𝜁𝑖 (𝑡)⟩ 𝑑𝑡

This implies

0

V (0) = ̃V0 ,

(144) 𝐻01

V0𝑚 󳨀→ V (0)

in

V0𝑚 󳨀→ V (0)


weakly,

𝑇

󸀠 + ∫ 𝛼 (𝑡) 𝑎 (V𝑚 (𝑡) , 𝑤𝑗 𝜁𝑖 (𝑡)) 𝑑𝑡 0

(145)

𝑇

+ ∫ 𝜇 (𝑡) 𝑎 (V𝑚 (𝑡) , 𝑤𝑗 𝜁𝑖 (𝑡)) 𝑑𝑡 0

From (115), we obtain

(154)

𝑇

+ ∫ ⟨𝑓 (V𝑚 (𝑡) + 𝜑 (𝑡)) , 𝑤𝑗 𝜁𝑖 (𝑡)⟩ 𝑑𝑡

⟨V𝑚 (0) , 𝑤𝑗 ⟩ = 𝜂 ⟨V𝑚 (𝑇) , 𝑤𝑗 ⟩

0

𝑇

󸀠 = 𝜂 (⟨V𝑚 (0) , 𝑤𝑗 ⟩ + ∫ ⟨V𝑚 (𝑡) , 𝑤𝑗 ⟩ 𝑑𝑡) ,

(146)

0

∀𝑗 ∈ N.

𝑇

0

= ∫ ⟨𝑓1 (𝑡) , 𝑤𝑗 𝜁𝑖 (𝑡)⟩ 𝑑𝑡, 0

for all 𝑖, 𝑗, 1 ≤ 𝑗 ≤ 𝑚, 𝑖 ∈ N. For 𝑖, 𝑗 fixed, passing to the limit in (154) by (140) and (153), we get

By (140), (145), and (146), it yields ⟨V (0) , 𝑤𝑗 ⟩ = 𝜂 (⟨V (0) , 𝑤𝑗 ⟩ + ∫ ⟨V󸀠 (𝑡) , 𝑤𝑗 ⟩ 𝑑𝑡)

𝑇

𝑇

(147)

= 𝜂 ⟨V (𝑇) , 𝑤𝑗 ⟩ , ∀𝑗 ∈ N.

∫ ⟨V󸀠 (𝑡) , 𝑤𝑗 𝜁𝑖 (𝑡)⟩ 𝑑𝑡 0

𝑇

+ ∫ 𝛼 (𝑡) 𝑎 (V󸀠 (𝑡) , 𝑤𝑗 𝜁𝑖 (𝑡)) 𝑑𝑡

Therefore,

0

V (0) = 𝜂V (𝑇) .

(148)

Therefore, (141) is proved. Using a compactness lemma ([18], Lions, p. 57), applied to (140), we can extract from the sequence {V𝑚 } a subsequence still denoted by {V𝑚 }, such that V𝑚 󳨀→ V


(149)

𝑇

+ ∫ 𝜇 (𝑡) 𝑎 (V (𝑡) , 𝑤𝑗 𝜁𝑖 (𝑡)) 𝑑𝑡 0

𝑇

+ ∫ ⟨𝑓 (V (𝑡) + 𝜑 (𝑡)) , 𝑤𝑗 𝜁𝑖 (𝑡)⟩ 𝑑𝑡 0

𝑇

= ∫ ⟨𝑓1 (𝑡) , 𝑤𝑗 𝜁𝑖 (𝑡)⟩ 𝑑𝑡. 0

(155)

16

Discrete Dynamics in Nature and Society Note that (155) holds for every 𝑖, 𝑗 ∈ N; that is, the equality 𝑇

𝑇

󸀠

On the other hand, ∫

󸀠

∫ ⟨V (𝑡) , 𝑤 (𝑡)⟩ 𝑑𝑡 + ∫ 𝛼 (𝑡) 𝑎 (V (𝑡) , 𝑤 (𝑡)) 𝑑𝑡 0

𝑇

0

0

= (1 − 𝜂2 ) ‖V (𝑇)‖20 ≥ 0;

𝑇

+ ∫ 𝜇 (𝑡) 𝑎 (V (𝑡) , 𝑤 (𝑡)) 𝑑𝑡 0

∫

(156)

𝑇

𝑇

0

+ ∫ ⟨𝑓 (V (𝑡) + 𝜑 (𝑡)) , 𝑤 (𝑡)⟩ 𝑑𝑡 𝑇

0

𝑑 󵄩2 󵄩 (𝛼 (𝑡) 󵄩󵄩󵄩V𝑥 (𝑡)󵄩󵄩󵄩0 ) 𝑑𝑡 𝑑𝑡

(160)

󵄩2 󵄩2 󵄩 󵄩 = 𝛼 (𝑡) 󵄩󵄩󵄩V𝑥 (𝑇)󵄩󵄩󵄩0 − 𝛼 (0) 󵄩󵄩󵄩V𝑥 (0)󵄩󵄩󵄩0

0

= ∫ ⟨𝑓1 (𝑡) , 𝑤 (𝑡)⟩ 𝑑𝑡,

𝑑 ‖V (𝑡)‖20 𝑑𝑡 = ‖V (𝑇)‖20 − ‖V (0)‖20 𝑑𝑡

󵄩2 󵄩 ≥ 𝛼 (0) (1 − 𝜂2 ) 󵄩󵄩󵄩V𝑥 (𝑇)󵄩󵄩󵄩0 ≥ 0.

∀𝑤 ∈ 𝐿2 (0, 𝑇; 𝐻01 ) , Hence, 𝑇

is fulfilled. Step 4 (uniqueness of the solutions). Let V1 and V2 be two solutions of (113). Then V = V1 − V2 satisfies the following problem: 𝑇

𝑇

0

0

𝑇

󵄩2 󵄩 2𝜇∗ ∫ 󵄩󵄩󵄩V𝑥 (𝑡)󵄩󵄩󵄩0 𝑑𝑡 ≤ −2 ∫ ⟨𝑓 (V1 (𝑡) + 𝜑 (𝑡)) 0 0 − 𝑓 (V2 (𝑡) + 𝜑 (𝑡)) , V (𝑡)⟩ 𝑑𝑡 𝑇

≤ 2𝛿 ∫ ‖V (𝑡)‖20 𝑑𝑡 ≤ 2𝛿 0

∫ ⟨V󸀠 (𝑡) , 𝑤 (𝑡)⟩ 𝑑𝑡 + ∫ 𝛼 (𝑡) 𝑎 (V󸀠 (𝑡) , 𝑤 (𝑡)) 𝑑𝑡

𝑅𝛾 (𝑅 − 1)2 2

(161)

𝑇

󵄩2 󵄩 ⋅ ∫ 󵄩󵄩󵄩V𝑥 (𝑡)󵄩󵄩󵄩0 𝑑𝑡. 0

𝑇

+ ∫ 𝜇 (𝑡) 𝑎 (V (𝑡) , 𝑤 (𝑡)) 𝑑𝑡

By 0 < 𝛿 < 2𝜇∗ /𝑅𝛾 (𝑅 − 1)2 , implying 𝛿(𝑅𝛾 /2)(𝑅 − 1)2
0, we have 𝑇

󵄩 󵄩2 ∫ (2𝜇 (𝑡) − 𝛼󸀠 (𝑡)) 󵄩󵄩󵄩V𝑥 (𝑡)󵄩󵄩󵄩0 𝑑𝑡 0

𝑇

󵄩2 󵄩 ≥ 2𝜇∗ ∫ 󵄩󵄩󵄩V𝑥 (𝑡)󵄩󵄩󵄩0 𝑑𝑡. 0

(159)

[1] A. B. Al’shin, M. O. Korpusov, and A. G. Sveshnikov, Blow-Up in Nonlinear Sobolev Type Equations, vol. 15 of De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co, Berlin, Germany, 2011. [2] R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, vol. 127 of Mathematics in Science and Engineering, Academic Press, Harcourt Brace Jovanovich, New York, NY, USA, 1976. [3] R. E. Showalter and T. W. Ting, “Asymptotic behavior of solutions of pseudo-parabolic partial differential equations,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 90, pp. 241– 258, 1971. [4] R. E. Showalter, “Existence and representation theorems for a semilinear Sobolev equation in Banach space,” SIAM Journal on Mathematical Analysis, vol. 3, pp. 527–543, 1972. [5] A. Mahmood, N. A. Khan, C. Fetecau, M. Jamil, and Q. Rubbab, “Exact analytic solutions for the flow of second grade fluid between two longitudinally oscillating cylinders,” Journal of Prime Research in Mathematics, vol. 5, pp. 192–204, 2009.

Discrete Dynamics in Nature and Society [6] S. Asghar, T. Hayat, and P. D. Ariel, “Unsteady Couette flows in a second grade fluid with variable material properties,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 1, pp. 154–159, 2009. [7] T. Hayat, M. Khan, A. M. Siddiqui, and S. Asghar, “Transient flows of a second grade fluid,” International Journal of NonLinear Mechanics, vol. 39, no. 10, pp. 1621–1633, 2004. [8] T. Hayat, M. Khan, M. Ayub, and A. M. Siddiqui, “The unsteady Couette flow of a second grade fluid in a layer of porous medium,” Archives of Mechanics, vol. 57, no. 5, pp. 405–416, 2005. [9] T. Hayat, M. Khan, and M. Ayub, “Some analytical solutions for second grade fluid flows for cylindrical geometries,” Mathematical and Computer Modelling, vol. 43, no. 1-2, pp. 16–29, 2006. [10] T. Hayat, M. Hussain, and M. Khan, “Hall effect on flows of an Oldroyd-B fluid through porous medium for cylindrical geometries,” Computers and Mathematics with Applications, vol. 52, no. 3-4, pp. 269–282, 2006. [11] T. Hayat, M. Sajid, and M. Ayub, “On explicit analytic solution for MHD pipe flow of a fourth grade fluid,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 4, pp. 745–751, 2008. [12] N. T. Long and A. P. N. Dinh, “On a nonlinear parabolic equation involving Bessel’s operator associated with a mixed inhomogeneous condition,” Journal of Computational and Applied Mathematics, vol. 196, no. 1, pp. 267–284, 2006. [13] L. T. P. Ngoc, N. Van Y, A. P. N. Dinh, and N. T. Long, “On a nonlinear heat equation associated with Dirichlet-Robin conditions,” Numerical Functional Analysis and Optimization, vol. 33, no. 2, pp. 166–189, 2012. [14] S. Cordier, L. X. Truong, N. T. Long, and A. P. N. Dinh, “Large time behavior of differential equations with drifted periodic coefficients modeling carbon storage in soil,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5641–5654, 2012. [15] L. T. P. Ngoc, T. T. Nhan, T. M. Thuyet, and N. T. Long, “On the nonlinear pseudoparabolic equation with the mixed inhomogeneous condition,” Boundary Value Problems, vol. 2016, article 137, 2016. [16] L. X. Truong, L. T. Ngoc, C. H. Hoa, and N. T. Long, “On a system of nonlinear wave equations associated with the helical flows of Maxwell fluid,” Nonlinear Analysis: Real World Applications, vol. 12, no. 6, pp. 3356–3372, 2011. [17] R. E. Showater, “Hilbert space methods for partial differential equations,” Electronic Journal of Differential Equations, Monograph 01, 1994. [18] J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Nonlinéaires, Dunod; Gauthier-Villars, Paris, France, 1969.

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