Abstract
The persistence properties of solutions to the dissipative 2component DegasperisProcesi system are investigated. We find that if the initial data with their derivatives of the system exponentially decay at infinity, then the corresponding solution also exponentially decays at infinity.
MSC: 35G25, 35L15, 35Q58.
Keywords:
2component; DegasperisProcesi system; dissipative; persistence properties1 Introduction
We consider the following dissipative 2component DegasperisProcesi system:
where λ, are nonnegative constants, , ().
In system (1.1), if , we get the classical DegasperisProcesi equation [1]
where represents the fluid velocity at time t in x direction and . The nonlinear convection term causes the steepening of wave form. The nonlinear dispersion effect term makes the wave form spread. The DegasperisProcesi equation has been studied in many works [28]. Escher et al.[2] demonstrated that there exists a unique solution to (1.2) with initial value (). Liu and Yin [3] obtained the global existence of solutions to (1.2). They derived several wave breaking mechanisms in Sobolev space with . Yin [4] established the local wellposedness for the DegasperisProcesi equation with initial value () on the line. In [5], the author obtained the global existence of solutions to the DegasperisProcesi equation on the circle. The precise blowup scenario was also derived. The global existence of strong solutions and global weak solutions to the DegasperisProcesi equation were shown in [6,7]. Guo et al.[9] studied the dissipative DegasperisProcesi equation,
where () is the dissipative term. They obtained the global weak solutions to (1.3). Guo [10] established the local wellposedness for (1.3), and also obtained the global existence, persistence properties and propagation speed of solutions. Wu and Yin [8] obtained the local wellposedness for (1.3), and also studied the blowup scenarios of solutions in periodic case.
On the other hand, many researchers have studied the integrable multicomponent generalizations of the DegasperisProcesi equation [1116]. Yan and Yin [11] investigated the 2component DegasperisProcesi system
where . They established the local wellposedness for system (1.4) in Besov space with , and also derived the precise blowup scenarios of strong solutions in Sobolev space with . Zhou et al.[12] investigated the traveling wave solutions to the 2component DegasperisProcesi system. Manwai [16] studied the selfsimilar solutions to the 2component DegasperisProcesi system. Fu and Qu [13] obtained the persistence properties of solutions to the 2component DegasperisProcesi system in Sobolev space with . For system (1.4), Jin and Guo [14] studied the blowup mechanisms and persistence properties of strong solutions.
Recently, a large amount of literature has been devoted to the study of the 2component CamassaHolm system [1728]. Hu [18] studied the dissipative periodic 2component CamassaHolm system
where . The author not only established the local wellposedness for system (1.5) in Besov space with , but she also presented global existence results and the exact blowup scenarios of strong solutions in Sobolev space with . For in system (1.5), Jin and Guo [19] considered the persistence properties of solutions to the modified 2component CamassaHolm system. Zhu [20] considered the persistence property of solutions to the coupled CamassaHolm system, and also established the global existence and blowup mechanisms of solutions. Guo [21,22] studied the persistence properties and unique continuation of solutions to the 2component CamassaHolm system in the case . It was shown in [29] that the dissipative CamassaHolm, DegasperisProcesi, HunterSaxton and Novikov equations could be reduced to the nondissipative versions by means of an exponentially timedependent scaling. One may refer to [3034] and the references therein for more details in this direction.
Motivated by the work in [13,20,35], we study the dissipative 2component DegasperisProcesi system (1.1). We note that the persistence properties of solutions to system (1.1) have not been discussed yet. The aim of this paper is to investigate the persistence properties of solutions in Sobolev space . The main idea of this work comes from [35].
Now we rewrite system (1.1) as
The main results are presented as follows.
Theorem 1.1Assumeandwith. Then the Cauchy problem (1.1) has a unique solution.
Theorem 1.2Letandwith. is the corresponding solution to system (1.1). If there existssuch that
then
Theorem 1.3Letandwith. is the corresponding solution to system (1.1). Assume the constant.
and there existssuch thatas, then
and there existssuch thatas, then
Theorem 1.4Letin system (1.1). Assumeandwith. is the corresponding solution to system (1.1). If there existssuch that
then
Theorem 1.5Assumeandwith. is the corresponding solution to system (1.1). If there existssuch that
then
Theorem 1.6Assumeandwith. is the corresponding solution to system (1.1). If there existssuch that
then
The remainder of this paper is organized as follows. In Section 2, the proofs of Theorems 1.1 and 1.2 are presented. Section 3 is devoted to the proofs of Theorems 1.3 and 1.4. The proofs of Theorems 1.5 and 1.6 are given in Section 4.
Notation We denote the norm of Lebesgue space , by , the norm in Sobolev space , by and the norm in Besov space , by . For , we denote
2 Proofs of Theorems 1.1 and 1.2
We write the definition of Besov space. One may check [3639] for more details.
Proposition 2.1[39]
Letand. The nonhomogeneous Besov space is defined by, where
2.1 Proof of Theorem 1.1
Using the LittlewoodPaley theory and estimates for solutions to the transport equation, one may follow similar arguments as in [11] to establish the local wellposedness for system (1.1) with some modification. Here we omit the detailed proof. For system (1.1) with initial data (), we see that the corresponding solution . Thus we complete the proof of Theorem 1.1.
2.2 Proof of Theorem 1.2
We denote
Multiplying the second equation in (1.6) by with and integrating the resultant equation with respect to x yield
We have
Thus
If , using the Sobolev embedding theorem, we have . Applying the Gronwall inequality to (2.2) yields
and, taking the limit as , we obtain
Multiplying the first equation in system (1.6) by with and integrating the resultant equation with respect to x yield
Using the Holder inequality, we have
which in combination with (2.3) yields
Using the Gronwall inequality, one derives
Taking the limit as in (2.5), one gets
Differentiating the first equation in (1.6) in the variable x yields
Multiplying (2.7) by with , integrating the resultant equation with respect to x and using
and
we have
We obtain
We introduce the weight function which is independent on t
where . It follows a.e. . Multiplying the first equation in system (1.6) and (2.7) by , we obtain
Multiplying (2.10) by and (2.11) by , respectively, and integrating the resultant equation with respect to x, we also note
As in the weightless case, we estimate and step by step as the previous estimates for u and . Thus
Multiplying the second equation in system (1.6) by , one deduces
Multiplying (2.13) by , integrating the resultant equation with respect to x and using
we have
Applying the Gronwall inequality and the Sobolev embedding theorem yields
Taking the limit as , one obtains
There exists which depends on , such that for all
Thus
Plugging (2.15), (2.16) into (2.12) and using (2.14), there exists such that
Using the Gronwall inequality, one deduces that for all and
Finally, taking the limit as , one obtains
Thus
uniformly on the interval . This completes the proof of Theorem 1.2.
3 Proofs of Theorems 1.3 and 1.4
3.1 Proof of Theorem 1.3
(1) For , integrating the first equation in (1.6) over the interval , one has
From the assumption in Theorem 1.3, one deduces
It follows from Theorem 1.2 that
For the right side in (3.1), we have
where . From Theorem 1.2, one has
Then
Noting , if there is at least one of the equalities and is valid, we have . Then there exists such that
Thus
which combined with the above estimates yields a contradiction. We obtain . Consequently, , .
(2) For , similar to the case , one deduces . Inserting into the second equation in (1.1), one derives
From (3.5), we have . This completes the proof of case (2) in Theorem 1.3.
3.2 Proof of Theorem 1.4
For , integrating the first equation in system (1.6) on the interval , one obtains
From the assumption in Theorem 1.4 as and Theorem 1.2, one deduces
For the right side in (3.6), firstly, we have
where . Using Theorem 1.2, we obtain
Thus
Noting
and
we have
Similarly, we have
Then as . From Theorem 1.2, if as , we have as . This completes the proof of Theorem 1.4.
4 Proofs of Theorems 1.5 and 1.6
4.1 Proof of Theorem 1.5
From the proof of Theorem 1.2, here we need to differentiate the second equation in (1.6) with variable x, and one has
Multiplying (4.1) by and integrating the resultant equation with respect to x, we also note
Thus, we obtain
Taking the limit as and applying the Gronwall inequality yield
In order to obtain the estimates for , we multiply (4.1) with the weight function , then
Multiplying (4.4) with and integrating the resultant equation with respect to x, we note
Hence, we have
Taking the limit as and using the Gronwall inequality, one obtains
From (4.6) and (2.17), one deduces that there exists such that
where
Applying the Gronwall inequality to (4.7), for all and , one has
Now taking the limit as in (4.8), one obtains
Using the assumption in Theorem 1.5, we complete the proof.
4.2 Proof of Theorem 1.6
The proof of Theorem 1.6 is similar to the proof of Theorem 1.3, here we omit it.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of this study equally and approved the final version of the manuscript.
Acknowledgements
The authors would like to express sincere gratitude to the anonymous referees for a number of valuable comments and suggestions. This work was partially supported by National Natural Science Foundation of P.R. China (71003082) and Fundamental Research Funds for the Central Universities (SWJTU12CX061 and SWJTU09ZT36).
References

Degasperis, A, Procesi, M: Asymptotic integrability. Symmetry and Perturbation Theory, pp. 23–37. World Scientific, Singapore (1999)

Escher, J, Liu, Y, Yin, Z: Global weak solutions and blowup structure for the DegasperisProcesi equation. J. Funct. Anal.. 241, 457–485 (2006). Publisher Full Text

Liu, Y, Yin, Z: Global existence and blowup phenomena for the DegasperisProcesi equation. Commun. Math. Phys.. 267, 801–820 (2006). Publisher Full Text

Yin, Z: On the Cauchy problem for an integrable equation with peakon solutions. Ill. J. Math.. 47, 649–666 (2003)

Yin, Z: Global existence for a new periodic integrable equation. J. Math. Anal. Appl.. 283, 129–139 (2003). Publisher Full Text

Yin, Z: Global solutions to a new integrable equation with peakons. Indiana Univ. Math. J.. 53, 1189–1210 (2004). Publisher Full Text

Yin, Z: Global weak solutions to a new periodic integrable equation with peakon solutions. J. Funct. Anal.. 212, 182–194 (2004). Publisher Full Text

Wu, S, Yin, Z: Blowup phenomena and decay for the periodic DegasperisProcesi equation with weak dissipation. J. Nonlinear Math. Phys.. 15, 28–49 (2008). Publisher Full Text

Guo, Y, Lai, S, Wang, Y: Global weak solutions to the weakly dissipative DegasperisProcesi equation. Nonlinear Anal.. 74, 4961–4973 (2011). Publisher Full Text

Guo, Z: Some properties of solutions to the weakly dissipative DegasperisProcesi equation. J. Differ. Equ.. 246, 4332–4344 (2009). Publisher Full Text

Yan, K, Yin, Z: On the Cauchy problem for a 2component DegasperisProcesi system. J. Differ. Equ.. 252, 2131–2159 (2012). Publisher Full Text

Zhou, J, Tian, L, Fan, X: Soliton, kink and antikink solutions of a 2component of the DegasperisProcesi equation. Nonlinear Anal., Real World Appl.. 11, 2529–2536 (2012)

Fu, Y, Qu, C: Unique continuation and persistence properties of solutions of the 2component DegasperisProcesi equations. Acta Math. Sci.. 32, 652–662 (2012). Publisher Full Text

Jin, L, Guo, Z: On a 2component DegasperisProcesi shallow water system. Nonlinear Anal., Real World Appl.. 11, 4164–4173 (2010). Publisher Full Text

Yu, L, Tian, L: Loop solutions, breaking kink wave solutions, solitary wave solutions and periodic wave solutions for the 2component DegasperisProcesi equation. Nonlinear Anal., Real World Appl.. 15, 140–148 (2014)

Manwai, Y: Selfsimilar blowup solutions to the 2component DegasperisProcesi shallow water system. Commun. Nonlinear Sci. Numer. Simul.. 16, 3463–3469 (2011). Publisher Full Text

Hu, Q: Global existence and blowup phenomena for a weakly dissipative 2component CamassaHolm system. Appl. Anal.. 92, 398–410 (2013). Publisher Full Text

Hu, Q: Global existence and blowup phenomena for a weakly dissipative periodic 2component CamassaHolm system. J. Math. Phys.. 52, (2011) Article ID 103701

Jin, L, Guo, Z: A note on a modified 2component CamassaHolm system. Nonlinear Anal., Real World Appl.. 13, 887–892 (2012). Publisher Full Text

Zhu, M: Blowup, global existence and persistence properties for the coupled CamassaHolm equations. Math. Phys. Anal. Geom.. 14, 197–209 (2011). Publisher Full Text

Guo, Z: Asymptotic profiles of solutions to the 2component CamassaHolm system. Nonlinear Anal.. 75, 1–6 (2012). Publisher Full Text

Guo, Z, Ni, L: Persistence properties and unique continuation of solutions to a 2component CamassaHolm equation. Math. Phys. Anal. Geom.. 14, 101–114 (2011). Publisher Full Text

Constantin, A, Ivanov, R: On an integrable 2component CamassaHolm shallow water system. Phys. Lett. A. 372, 7129–7132 (2008). Publisher Full Text

Gui, G, Liu, Y: On the global existence and wavebreaking criteria for the 2component CamassaHolm system. J. Funct. Anal.. 258, 4251–4278 (2010). Publisher Full Text

Gui, G, Liu, Y: On the Cauchy problem for the 2component CamassaHolm system. Math. Z.. 268, 45–66 (2011). Publisher Full Text

Ai, X, Gui, G: Global wellposedness for the Cauchy problem of the viscous DegasperisProcesi equation. J. Math. Anal. Appl.. 361, 457–465 (2010). Publisher Full Text

Chen, W, Tian, L, Deng, X, Zhang, J: Wave breaking for a generalized weakly dissipative 2component CamassaHolm system. J. Math. Anal. Appl.. 400, 406–417 (2013). Publisher Full Text

Jin, L, Jiang, Z: Wave breaking of an integrable CamassaHolm system with 2component. Nonlinear Anal.. 95, 107–116 (2014)

Lenells, J, Wunsch, M: On the weakly dissipative CamassaHolm, DegasperisProcesi, and Novikov equations. J. Differ. Equ.. 255, 441–448 (2013). Publisher Full Text

Zhu, M: On a shallow water equation perturbed in Schwartz class. Math. Phys. Anal. Geom. (2012). Publisher Full Text

Zhu, M, Jiang, Z: Some properties of solutions to the weakly dissipative bfamily equation. Nonlinear Anal., Real World Appl.. 13, 158–167 (2012). Publisher Full Text

Ni, L, Zhou, Y: Wellposedness and persistence properties for the Novikov equation. J. Differ. Equ.. 250, 3002–3021 (2011). Publisher Full Text

Zong, X: Properties of the solutions to the 2component bfamily systems. Nonlinear Anal.. 75, 6250–6259 (2012). Publisher Full Text

Zhou, S, Mu, C: The properties of solutions for a generalized bfamily equation with peakons. J. Nonlinear Sci.. 23, 863–889 (2013). Publisher Full Text

Himonas, A, Misiolek, G, Ponce, G, Zhou, Y: Persistence properties and unique continuation of solutions to the CamassaHolm equation. Math. Phys. Anal. Geom.. 271, 511–522 (2007)

Danchin, R: A few remarks on the CamassaHolm equation. Differ. Integral Equ.. 14, 953–988 (2001)

Danchin, R: A note on wellposedness for CamassaHolm equation. J. Differ. Equ.. 192, 429–444 (2003). Publisher Full Text

Danchin, R: Fourier analysis methods for PDEs. Lecture Notes, 14 Nov. Preprint (2005)

Bahouri, H, Chemin, J, Danchin, R: Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin (2010)