Abstract
We are concerned with the global existence of classical solutions for a general model of viscosity long-short wave equations. Under suitable initial conditions, the existence of the global classical solutions for the viscosity long-short wave equations is proved. If it does not exist globally, the life span which is the largest time where the solutions exist is also obtained.
1. Introduction
In this paper, we studied the global well-posedness of the solutions for the long-short wave systems with viscosity which describes the coupling between nonlinear Schrödinger systems and the parabolical systems. In 1977, Benney [2] presented a general theory for deriving nonlinear partial differential equations in which both long and short wave solutions coexist and interact with each other nonlinearly.where denotes the envelope of the short wave, is the amplitude of the long wave, the quantity is the long wave speed, and is the group velocity of the short waves. This system arises in the study of surface waves with both gravity and capillary models presented [8] and also in plasma physics [21].
In [20], Tsutsumi and Hatano studied the well-posedness of the Cauchy problems for one type of Benney equationsand they got the well-posedness of the Cauchy problems of (2).
In [5, 6], the authors investigated the quasilinear Benney equations with the form,
In [5], they imposed the condition and they established the existence of weak solutions for equation (3). In [6], they supposed that is a polynomial real function, and they got the existence of local strong solutions for these versions of Benney equations. However, for the existence of the smooth solutions associated with an arbitrary flux-function and the finite-time blow-up of these equations are still open. There are many papers that deal with the long-short wave equations, for example [1, 4, 14, 15, 17, 22, 23].
In the present paper, we extended the results to the viscosity long-short wave equations of the general form,where , , and are nonlinear functions satisfying hypotheses: in a neighborhood of , say, for , and where is an integer , , and is the viscosity coefficient.
Precisely, we are concerned with the viscosity long-short wave equations with the following form:
Our purpose is to estimate the life span of solutions which is expressed explicitly in [11, 16].We call the maximal existence time of the solutions in the classical sense the life span (or blow-up time) of . We denote by for the life span of the solutions of (5). Under the same assumptions on and as above, we will give an upper estimate of in terms of the prescribed conditions and study the best possibility of this estimate. Thus, we can get the existence of the global classical solutions for the viscosity long-short wave equations. If it does not exist globally, we can also get the life span (the largest time in which the solutions exist) with a more general in .The studies about the life span are bound in literatures [3, 7, 9, 10, 12, 13, 18, 19]. Recently, some new results about these problems are founded in [24–36].
We state our main results as follows.
Theorem 1. Suppose that the nonlinear term on the right-hand side of (5) satisfies (43) and
Then, for any given integer , there exist positive constants and with such that, for any , there exists a positive number such that Cauchy problem (5) admits on a unique classical solution , where can be chosen as follows:where are positive constants satisfying (148), (149), and (151). Moreover, with eventual modification on a set with zero measure in the variable , we can obtain the results; for any finite with , we have
Moreover, by the Sobolev embedding theorems (observing that ), it easily follows from (8)–(11) that the solutions are classical solutions to the Cauchy problem (5).
Remark 1. In this paper, we took the viscosity coefficient for simplicity. For the case when , we will discuss it in another paper.
The remainder of this paper is as follows: Section 2 is devoted to giving some useful estimates which will be used in Section 3. In Section 3, we consider the Cauchy problem for n-dimensional nonlinear evolution equations and we get our main results. In Section 4, conclusion is given.
2. Preliminary
First, we considered the following homogeneous equations:
It is well known that, by means of the Fourier transformation, the solutions to the Cauchy problem of equations (12) and (13) can be expressed in the following explicit form:andwhere and .
For simplicity, we wrote (14) in the form,and (15) in the form,where
Then, by Duhamel’s principle, the solutions to the Cauchy problem for inhomogeneous heat equationscan be denoted asandor precisely,
Now, we used the explicit expressions (14) and (15) to establish some decay estimates for solutions to Cauchy problem (12) and (13) for the n-dimensional homogeneous equations.
The following lemmas were employed by Li and Chen [16], but for completeness, we included them here without proving.
Lemma 1. Suppose that all norms appearing on the right-hand side below are bounded. For any integer , solutions (14) and (15) to the Cauchy problem (12) and (13) satisfy the following estimates:and
Lemma 2. Suppose that , where is an integer, then the Cauchy problemadmits the following estimate:where is a positive constant independent of is a multi-index,, and .
Lemma 3. Under assumption , if all norms appearing on the right-hand side below are bounded, then for any given integer , we have
Furthermore, suppose that is a sufficiently smooth function of , satisfying that ifthen
For any given integer , if a vector function satisfiesand such that all norms appearing on the right-hand side below are bounded, thenwhere is a positive constant (depending on ) and
In particular, we havewhere is a positive constant (depending on ) and
Lemma 4. Suppose that satisfies the same assumptions as in Lemma 3. If vector functions and satisfy (30), respectively, and such that all norms appearing on the right-hand side below are bounded, then for any given integer , let
We havewhere and satisfy and is a positive constant (depending on ).
3. The Proof of Theorem 1
In this section, we shall discuss the Cauchy problem for n-dimensional nonlinear evolution equations:where , and is a small parameter.
Let
For the nonlinear term in (39), we give the following hypotheses: in a neighborhood of , say, for is sufficiently smooth andwhere is an integer .
For any given integer such that , and any positive numbers , we introduce the following set of functions:where
We now define a mapwhere are the solutions of the linear equations
The next step is to prove that the map is a strict contraction in the complete metric space .
Lemma 5. For any (in which ), satisfies
Proof. From (21), we have
By (23), we have
A combination of (50)–(52) leads to
Noting that , by the definition of , we have
On combining (53)–(57), we get
We point out the fact that, if , then we have
The proof of (59) was shown in [16], but to complete it, we include here. Let
The combination of (60) and (61) gives (59).
Thus, from (58) and (59), we obtain
By (49) and (24), we also have
Moreover, in fact,
Thus, we have
On combining (63)–(65) and (67) and the definition of , we obtain
By (28), we have
Noting the hypotheses of (43), it follows from (33) that
Using (67), , and the definition of , we obtain
We can also get
By (25), we have
From (67), we have
Thus,
n deriving (76), we have used .
Using (29), we haveand
On combining (73)–(78), we obtain
Thus,
Finally, from (26) and (72), we have
On combining (75) and (81), we obtain
On combining the above discussions, we obtain
Lemma 6. For any , (in which and , if also satisfy , thenwhere .
Proof. By the definition of , we have
Similar to (53), we have
Noting that , and the definition of and using the hypotheses (43) and (38) (in which we take ), we get
For we havewe also have
Thus, we can get
Putting (88) and (91) into (87), and using the definition of and (59), we obtain
Next, similar to (63), we have
Using (38) (in which we take r = 1, p = q = 2), we have
By the definition of and the definition of , we get
Moreover, we have
Thus, we see
Similarly, we have
Besides, we have
On combining (94)–(101), we obtain
Noting that is an integer , we have
It follows from (102) that
Similar to (91), we can get
On combining (93), (104), and (105), we get
By means of (29), it comes from (85) that
By (38) (in which we take r = q = 2 and ), we get
Still using (97), (99), and (101) and notifying thatit follows that
For we know that
Thus,in deriving (111), we have used and .
On combining (107), (111), and (112) and using , we obtain
Similar to (73), we obtain
We have from (38) that
We can obtainand we also have
For we have
Using (99), we obtain
So, we have
Following the same procedure as (122), we obtain
We now begin to deal with .
We can easily get
Thus, we have
Now, we need to seek a new method to give a bound for
For we know that,is a continuous differentiable function in , thus we have
We can easily get when , we have .
For we have
Thus, we obtain
So, we can easily see that
On combining (125)–(132), we obtain
Using (38), we obtain
On combining (116)–(134), we obtain
Finally, applying (26), we have
From (38) (taking ), we get
Thus, we can easily get
Similar to (119), we also have
We can get
We need only to give a bound for .
We obtain
On combining (135)–(144), (97), and (99), we can conclude that
On combining (92), (106), (115), (135), and (145), we obtain (84).
Lemma 7. Let , where are the constants given in Lemma 5 and Lemma 6, respectively. If there exists a positive number with such that for any , if and allowthen the map has a unique fixed point in .
Proof. By (146) and Lemmas 5 and 6, it is easy to see that for any satisfiesand that for any ,Namely, maps into itself. Moreover, is a contraction with respect to the metric . Thus, it follows from the standard contracting mapping principle that the map possesses a fixed point . The proof of Lemma 7 is finished.
The fixed point obtained in Lemma 7 is obviously the classical solution to the Cauchy problem (39)–(41) on .
Now, we determine and for any such that (145) holds. In what follows, we always take so small that .(1)In the case that , since(i)We can choose , and let be so small that for any , (145) holds. In this case, we get the global solution.(2)In the case that , since(i)we can choose , where is a positive constant that satisfies(ii)Thus, if is so small that(145) holds. In that case, we get the so-called almost global solutions.(3)In the case that , sincewe can choose , where is a positive constant that satisfiesThus, we still get (145), provided (153) holds.
On combining the above discussions, we get the desired Theorem 1.
4. Conclusion
In the present paper, the global existence and blow-up for the classical solutions of the long-short wave equations with viscosity are studied. Using the method proposed in this paper, similar results can be obtained for other wave equations with different nonlinear terms. Our method is also valid for the weak solution. For the case when , our method is no longer applicable. There may be difficulty in obtaining the energy estimates. We must seek new method to overcome this difficulty. We think it is interesting. We will discuss it in another paper.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.