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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 568404, 12 pages
The Local Strong and Weak Solutions for a Generalized Pseudoparabolic Equation
Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China
Received 7 February 2012; Accepted 21 March 2012
Academic Editor: Yonghong Wu
Copyright © 2012 Nan Li. 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.
The Cauchy problem for a nonlinear generalized pseudoparabolic equation is investigated. The well-posedness of local strong solutions for the problem is established in the Sobolev space with , while the existence of local weak solutions is proved in the space with . Further, under certain assumptions of the nonlinear terms in the equation, it is shown that there exists a unique global strong solution to the problem in the space with .
Davis  investigated the pseudoparabolic equation where the constant , the function , and , and the subscripts and indicate partial derivatives. Equation (1.1) arises from the study of shearing flows of incompressible simple fluids. The quantity is viewed as an approximation to the stress functional during such a flow. Much attention has been given to this approximation when the function 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 .
Recently, Chen and Xue  investigated the Cauchy problem for the nonlinear generalized pseudoparabolic equation where is an unknown function, , , is a real number, , , and 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  in which several key assumptions are imposed on the functions and . In fact, various dynamic properties for many special cases of (1.2) have been established in [5####^~^~^~^~^~^####x2013;7]. For example, when , (1.2) becomes the generalized regularized long wave Burger equation.
Motivated by the works in [1, 4], we study the problem where and , is a nature number, is a given function, and 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 . Letting , 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 with . We should address that the Sobolev index is required to guarantee the local well-posedness of (1.1) and (1.2) in the works of Davis  and Chen and Xue . 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 and the initial value similar to those presented in [1, 4], problem (1.3) is shown to have a unique global solution in the space .
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.
2. Local Well-Posedness
Let ####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009; be the space of all measurable functions such that . We define with the standard norm . For any real number , denotes the Sobolev space with the norm defined by where .
For and nonnegative number , denotes the Frechet space of all continuous -valued functions on . We set . For simplicity, throughout this paper, we let denote any positive constants.
The local well-posedness theorem is stated as follows.
Theorem 2.1. Provided that , , is a polynomial of order with . Then problem (1.3) admits a unique local solution:
Proof. In fact, the first equation of problem (1.3) is equivalent to the equation
which leads to
Suppose that both and are in the closed ball of radius about the zero function in and is the operator in the right-hand side of (2.4), for fixed , we get
The algebraic property of with (see [8####^~^~^~^~^~^####x2013;10]) and derives that
Using and , we get
in which is used.
From (2.5)####^~^~^~^~^~^####x2013;(2.7), we obtain where and is independent of . Choosing sufficiently small such that , we know that is a contractive mapping. Applying the above inequality and (2.4) yields Choosing sufficiently small such that , we know that maps to itself. It follows from the contractive mapping principle that the mapping has a unique fixed point in . This completes the proof of Theorem 2.1.
3. Existence of Local Weak Solutions
In this section, we assume that where 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 ). If , then is an algebra. Moreover, where is a constant depending only on .
Lemma 3.2 (see Kato and Ponce ). Let . If and , then
Lemma 3.3. Let , , and the function is a solution of problem (1.3) and the initial data . Then the following results hold.
For , there is a constant such that
For , there is a constant such that
Proof. For , applying to both sides of the first equation of system (1.3) and integrating with respect to 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
For the second term, we have
For , applying Lemma 3.1 derives
For , we get
It follows from (3.5)####^~^~^~^~^~^####x2013;(3.9) that there exists a constant such that Integrating both sides of the above inequality with respect to results in inequality (3.3).
To estimate the norm of , we apply the operator to both sides of the first equation of system (1.3) to obtain the equation Applying to both sides of (3.11) for gives rise to For the right-hand of (3.12), we have
Applying (3.13) into (3.12) yields the inequality for a constant . This completes the proof of Lemma 3.3.
Lemma 3.4. If is a solution of problem (1.3), , , then where is a constant.
Proof. Multiplying both sides of the first equation of (1.3) by and integrating with respect to over , we have
Since we derive that which results in
From (3.19), we know that (3.15) holds. This completes the proof.
Defining and setting with and , we know that for any and .
It follows from Theorem 2.1 that for each the Cauchy problem has a unique solution .
Lemma 3.5. Under the assumptions of problem (3.21), the following estimates hold for any with , and where is a constant independent of .
Proof. Using the definition of and results in the conclusion of the lemma.
Lemma 3.6. Suppose that with such that . Let be defined as in system (3.21) and let . Then there exist two positive constants and , independent of , such that the solution of problem (3.21) satisfies for any .
Proof. Using notation and differentiating both sides of the first equation of problem (3.11) with respect to give rise to
Letting be an integer and multiplying the above equation by and then integrating the resulting equation with respect to yield the equality
Applying the H####^~^~^~^~^~^####xf6;lder####^~^~^~^~^~^####x2019;s inequality to (3.24) and noting Lemmas 3.4 and 3.5, we obtain or
Since as for any , integrating both sides of the inequality (3.27) with respect to and taking the limit as result in the estimate Using the algebra property of with yields ( means that there exists a sufficiently small such that ) in which Lemmas 3.4 and 3.5 are used. From (3.28) and (3.29), one has
From Lemma 3.5, it follows from the contraction mapping principle that there is a such that the equation has a unique solution . Using the result presented on page####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;51 in  yields that there are constants and independent of such that for arbitrary , which leads to the conclusion of Lemma 3.6.
Using Lemmas 3.3####^~^~^~^~^~^####x2013;3.6, notation and Gronwall####^~^~^~^~^~^####x2019;s inequality result in the inequalities where , and depends on . It follows from the Aubin####^~^~^~^~^~^####x2019;s compactness theorem that there is a subsequence of , denoted by , such that and their temporal derivatives are weakly convergent to a function and its derivative in and , respectively. Moreover, for any real number , is convergent to the function strongly in the space and converges to strongly in the space for . Thus, we can prove the existence of a weak solution to (1.3).
Theorem 3.7. Suppose that with , and . Then there exists a such that (1.3) subject to initial value has a weak solution in the sense of distribution and .
Proof. From Lemma 3.6, we know that is bounded in the space . Thus, the sequences , , , and are weakly convergent to , , , and in for any , separately. Therefore, satisfies the equation with and . Since is a separable Banach space and is a bounded sequence in the dual space of , there exists a subsequence of , still denoted by , weakly star convergent to a function in . It derives from the weakly convergence of to in that almost everywhere. Thus, we obtain .
4. Well-Posedness of Global Solutions
Lemma 4.1. If is a solution of problem (1.3), , , then where
Proof. Multiplying each side of the first equation of problem (1.3) by and integrating over yields Integrating the right-hand side of the above identity by parts and using , we get From (4.3), (4.4) and the assumption of this lemma, we have from which we obtain (4.1).
Theorem 4.2. Suppose that , , with positive integer . Then problem (1.3) has a unique global solution:
- P. L. Davis, “A quasilinear parabolic and a related third order problem,” Journal of Mathematical Analysis and Applications, vol. 40, pp. 327–335, 1972.
- B. D. Coleman and W. Noll, “An approximation theorem for functionals, with applications in continuum mechanics,” Archive for Rational Mechanics and Analysis, vol. 6, no. 1, pp. 355–370, 1960.
- R. E. Showalter and T. W. Ting, “Pseudoparabolic partial differential equations,” SIAM Journal on Mathematical Analysis, vol. 1, pp. 1–26, 1970.
- G. W. Chen and H. X. Xue, “Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation,” Journal of Differential Equations, vol. 245, no. 10, pp. 2705–2722, 2008.
- P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop, and E. S. Titi, “Continuum approach to discreteness,” Physical Review E, vol. 65, no. 4, Article ID 046613, 13 pages, 2002.
- J. L. Bona and L. Luo, “Asymptotic decomposition of nonlinear, dispersive wave equations with dissipation,” Physica D, vol. 152–153, pp. 363–383, 2001.
- J. Albert, “Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation,” Journal of Differential Equations, vol. 63, no. 1, pp. 117–134, 1986.
- T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988.
- S. Y. Lai and Y. H. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,” Journal of Differential Equations, vol. 248, no. 8, pp. 2038–2063, 2010.
- S. Y. Lai and Y. H. Wu, “Existence of weak solutions in lower order Sobolev space for a Camassa-Holm-type equation,” Journal of Physics A, vol. 43, no. 9, Article ID 095205, 13 pages, 2010.
- Y. A. Li and P. Olver, “Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,” Journal of Differential Equations, vol. 162, no. 1, pp. 27–63, 2000.