The Local Strong and Weak Solutions for a Generalized Pseudoparabolic Equation

and Applied Analysis 3 For T > 0 and nonnegative number s, C 0, T ;H R denotes the Frechet space of all continuous H-valued functions on 0, T . We set Λ 1 − ∂x . For simplicity, throughout this paper, we let c denote any positive constants. The local well-posedness theorem is stated as follows. Theorem 2.1. Provided that s ≥ 3/2, u0 ∈ H R , φ is a polynomial of order N with φ 0 0. Then problem 1.3 admits a unique local solution: u t, x ∈ C 0, T ;H R ⋂ C1 ( 0, T ;Hs−1 R ) . 2.2 Proof. In fact, the first equation of problem 1.3 is equivalent to the equation ut Λ−2 ( ∂ ∂x φ ux βuuxx ) , 2.3


Introduction
Davis 1 investigated the pseudoparabolic equation where the constant α ≥ 0, the function ϕ ∈ C 2 −∞, ∞ , ϕ 0 0 and ϕ ξ > 0, and the subscripts x and t indicate partial derivatives.Equation 1.1 arises from the study of shearing flows of incompressible simple fluids.The quantity ϕ u x αu tx is viewed as an approximation to the stress functional during such a flow.Much attention has been given to this approximation when the function ϕ is linear see 2, 3 .The existence and uniqueness of the global weak solution of the initial value problem for 1.1 were established in 1 .
Recently, Chen and Xue 4 investigated the Cauchy problem for the nonlinear generalized pseudoparabolic equation where u t, x is an unknown function, α > 0, λ ≥ 0, γ is a real number, f s , ϕ s , and g s denote given nonlinear functions.The well-posedness of global strong solution in a Sobolev space, the global classical solution and its asymptotic behavior are studied in 4 in which several key assumptions are imposed on the functions ϕ s and g s .In fact, various dynamic properties for many special cases of 1.2 have been established in 5-7 .For example, when ϕ s g s 0, 1.2 becomes the generalized regularized long wave Burger equation.Motivated by the works in 1, 4 , we study the problem where α > 0 and β ≥ 0, m is a nature number, ϕ s is a given function, and u 0 x is a given initial value function.Here we should address that 1.2 does not include the first equation of problem 1.3 due to the term βu 2m u xx .Letting β 0, the first equation of problem 1.3 reduces to 1.1 .The objectives of this work are threefold.The first objective is to establish the local well-posedness of system 1.3 in the space C 0, T ; H s R C 1 0, T ; H s−1 R with s > 3/2.We should address that the Sobolev index s ≥ 2 is required to guarantee the local wellposedness of 1.1 and 1.2 in the works of Davis 1 and Chen and Xue 4 .The second aim is to study the existence of local weak solutions for system 1.3 .The third aim is to discuss the well-posedness of the global strong solution for problem 1.3 .Under the assumptions of the function ϕ s and the initial value u 0 x similar to those presented in 1, 4 , problem 1.3 is shown to have a unique global solution in the space C 0, ∞ ; H s R C 1 0, ∞ ; H s−1 R .The organization of this paper is as follows.The well-posedness of local strong solutions for problem 1.3 is investigated in Section 2, and the existence of local weak solutions is established in Section 3. Section 4 deals with the well-posedness of the global strong solution.

Local Well-Posedness
For any real number s, H s H s R denotes the Sobolev space with the norm defined by where h t, ξ R e −ixξ h t, x dx.
For T > 0 and nonnegative number s, C 0, T ; H s R denotes the Frechet space of all continuous H s -valued functions on 0, T .We set Λ 1 − ∂ 2 x 1/2 .For simplicity, throughout this paper, we let c denote any positive constants.The local well-posedness theorem is stated as follows.
Theorem 2.1.Provided that s ≥ 3/2, u 0 ∈ H s R , ϕ is a polynomial of order N with ϕ 0 0. Then problem 1.3 admits a unique local solution: Proof.In fact, the first equation of problem 1.3 is equivalent to the equation Suppose that both u and v are in the closed ball B M 0 0 of radius M 0 > 1 about the zero function in C 0, T ; H s R and A is the operator in the right-hand side of 2.4 , for fixed t ∈ 0, T , we get

2.5
The algebraic property of H s 0 R with s 0 > 1/2 see 8-10 and s > 3/2 derives that

2.6
Using in which s > 3/2 is used.From 2.5 -2.7 , we obtain where θ max cT M N−1 0 , cTM 2m 0 and c is independent of T .Choosing T sufficiently small such that θ < 1, we know that A is a contractive mapping.Applying the above inequality and 2.4 yields 2.9 Choosing T sufficiently small such that θM 0 u 0 H s < M 0 , we know that A maps B M 0 0 to itself.It follows from the contractive mapping principle that the mapping A has a unique fixed point u in B M 0 0 .This completes the proof of Theorem 2.1.

Existence of Local Weak Solutions
In this section, we assume that ϕ η η 2N 1 where N is a nature number.In order to establish the existence of local weak solution, we need the following lemmas.

Lemma 3.1 see Kato and Ponce
where c is a constant depending only on r.

Lemma 3.2 see Kato and Ponce
x , and the function u t, x is a solution of problem 1.3 and the initial data u 0 x ∈ H s .Then the following results hold.
For q ∈ 0, s − 1 , there is a constant c such that

3.3
For q ∈ 0, s − 1 , there is a constant c such that Proof.For q ∈ 0, s − 1 , applying Λ q u Λ q to both sides of the first equation of system 1.3 and integrating with respect to x by parts, we have the identity We will estimate the two terms on the right-hand side of 3.5 , respectively.For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.1 and 3.2, we have

3.6
For the second term, we have

3.7
For K 1 , applying Lemma 3.1 derives For K 2 , we get 3.9 It follows from 3.5 -3.9 that there exists a constant c such that 1 2

3.10
Integrating both sides of the above inequality with respect to t results in inequality 3.3 .
To estimate the norm of u t , we apply the operator 1 − ∂ 2 x −1 to both sides of the first equation of system 1.3 to obtain the equation

3.11
Applying Λ q u t Λ q to both sides of 3.11 for q ∈ 0, s − 1 gives rise to R Λ q u t 2 dx R Λ q u t Λ q−2 ∂ x ϕ u x u 2m u xx dτ.

3.12
For the right-hand of 3.12 , we have

3.13
Abstract and Applied Analysis 7 Applying 3.13 into 3.12 yields the inequality 3.14 for a constant c > 0. This completes the proof of Lemma 3.3.
where c is a constant.
Proof.Multiplying both sides of the first equation of 1.3 by u t, x and integrating with respect to x over R, we have x dx < 0, 3.17 we derive that 1 2
It follows from Theorem 2.1 that for each ε the Cauchy problem Lemma 3.5.Under the assumptions of problem 3.21 , the following estimates hold for any ε with 0 < ε < 1/4, u 0 ∈ H s R and s > 0 : where c 1 is a constant independent of ε.
Proof.Using the definition of u ε0 and u ε0x results in the conclusion of the lemma.
Let u ε0 be defined as in system 3.21 and let ϕ η η 2N 1 .Then there exist two positive constants T and c, independent of ε, such that the solution u ε of problem 3.21 satisfies u εx L ∞ ≤ c for any t ∈ 0, T .
Proof.Using notation u u ε and differentiating both sides of the first equation of problem 3.11 with respect to x give rise to

3.23
Letting p > 0 be an integer and multiplying the above equation by u x 2p 1 and then integrating the resulting equation with respect to x yield the equality where

3.25
Applying the H ölder's inequality to 3.24 and noting Lemmas 3.4 and 3.5, we obtain

3.27
Since , integrating both sides of the inequality 3.27 with respect to t and taking the limit as p → ∞ result in the estimate Using the algebra property of H s 0 R with s 0 > 1/2 yields u ε H 1/2 means that there exists a sufficiently small δ > 0 such that

3.29
in which Lemmas 3.4 and 3.5 are used.From 3.28 and 3.29 , one has

3.30
From Lemma 3.5, it follows from the contraction mapping principle that there is a T > 0 such that the equation has a unique solution W ∈ C 0, T .Using the result presented on page 51 in 11 yields that there are constants T > 0 and c > 0 independent of ε such that u x L ∞ ≤ W t L ∞ ≤ c for arbitrary t ∈ 0, T , which leads to the conclusion of Lemma 3.6.
Using Lemmas 3.3-3.6,notation u ε u and Gronwall's inequality result in the inequalities

3.32
where q ∈ 0, s , r ∈ 0, s − 1 1 ≤ s ≤ 3/2 and C T depends on T .It follows from the Aubin's compactness theorem that there is a subsequence of {u ε }, denoted by {u ε n }, such that {u ε n } and their temporal derivatives {u ε n t } are weakly convergent to a function u t, x and its derivative u t in L 2 0, T , H s and L 2 0, T , H s−1 , respectively.Moreover, for any real number R 1 > 0, {u ε n } is convergent to the function u strongly in the space L 2 0, T , H q −R 1 , R 1 and {u ε n t } converges to u t strongly in the space L 2 0, T , H r −R 1 , R 1 for r ∈ 0, s − 1 .Thus, we can prove the existence of a weak solution to 1.3 .Theorem 3.7.Suppose that u 0 x ∈ H s with 1 ≤ s ≤ 3/2, u 0x L ∞ < ∞ and ϕ η η 2N 1 .Then there exists a T > 0 such that 1.3 subject to initial value u 0 x has a weak solution u t, x ∈ L 2 0, T , H s in the sense of distribution and u x ∈ L ∞ 0, T × R .
Proof.From Lemma 3.6, we know that {u ε n x } ε n → 0 is bounded in the space L ∞ .Thus, the sequences {u ε n }, {u ε n x }, {u 2 ε n x }, and {u 2N 1 ε n x } are weakly convergent to u, u x , u 2 x , and u 2N 1 x in L 2 0, T , H r −R, R for any r ∈ 0, s − 1 , separately.Therefore, u satisfies the equation x g dx dt, 3.33 with u 0, x u 0 x and g ∈ C ∞ 0 .Since X L 1 0, T × R is a separable Banach space and {u ε n x } is a bounded sequence in the dual space X * L ∞ 0, T × R of X, there exists a subsequence of {u ε n x }, still denoted by {u ε n x }, weakly star convergent to a function v in L ∞ 0, T × R .It derives from the weakly convergence of {u ε n x } to u x in L 2 0, T × R that u x v almost everywhere.Thus, we obtain u x ∈ L ∞ 0, T × R .

Well-Posedness of Global Solutions
2m u x g x − 2mβu 2m−1 u 2

Lemma 4 . 1 .Proof.
If u t, x is a solution of problem 1.3 , α > 0, ϕ η η 2N 1 , then u x L ∞ ≤ A 1/2 ,Multiplying each side of the first equation of problem 1.3 by u xx and integrating over 0, t × R yields -hand side of the above identity by parts and using u x ±∞ 0