Global Existence and Blow-Up for the Classical Solutions of the Long-Short Wave Equations with Viscosity

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.


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 u(t, x) denotes the envelope of the short wave, v(t, x) is the amplitude of the long wave, the quantity C l is the long wave speed, and C g is the group velocity of the short waves.
is 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 equations iu t + △u � vu +|u| 2 u, v t � |u| 2 x , ⎧ ⎨ ⎩ (2) and 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 f(v) � av 2 − bv 3 , a > 0, b > 0 and they established the existence of weak solutions for equation (3). In [6], they supposed that f 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 fluxfunction f and the finite-time blow-up of these equations are still open. ere are many papers that deal with the long-short wave equations, for example [1,4,14,15,17,22,23].
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 (u, v) in the classical sense the life span (or blow-up time) of (u, v). We denote by T(ε) for ε > 0 the life span of the solutions (u, v) of (5). Under the same assumptions on ϕ and g as above, we will give an upper estimate of T(ε) in terms of the prescribed conditions and study the best possibility of this estimate. us, 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 f(v) in R n . e 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][25][26][27][28][29][30][31][32][33][34][35][36].
We state our main results as follows.
Theorem 1. Suppose that the nonlinear term f on the righthand side of (5) satisfies (43) and en, for any given integer S ≥ n + 5, there exist positive constants ε 0 and c 0 with c 0 ε 0 ≤ 1 such that, for any ε ∈ (0, ε 0 ], there exists a positive number T � T(ε) such that Cauchy problem (5) admits on [0, T(ε)] a unique classical solution u ∈ X S,c 0 ε,T(ε) , where T(ε) can be chosen as follows: where a, b are positive constants satisfying (148), (149), and (151). Moreover, with eventual modification on a set with zero measure in the variable t, we can obtain the results; for any finite T 0 with 0 < T 0 ≤ T, we have Moreover, by the Sobolev embedding theorems (observing that S ≥ n + 5), 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 η � 1 for simplicity. For the case when η ⟶ 0, we will discuss it in another paper. e 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.

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: and v(x, t) � 1 (4πt) n/2 R n e − |x− y| 2 /4t g(y)dy, (15) where y � (y 1 , y 2 , . . . , y n ) and |x − y| 2 � n i�1 (x i − y i ) 2 . For simplicity, we wrote (14) in the form, and (15) in the form, where en, by Duhamel's principle, the solutions to the Cauchy problem for inhomogeneous heat equations can be denoted as 2 Discrete Dynamics in Nature and Society or precisely, Now, we used the explicit expressions (14) and (15) to establish some decay estimates t ⟶ ∞ for solutions to Cauchy problem (12) and (13) for the n-dimensional homogeneous equations. e 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 righthand side below are bounded. For any integer N ≥ 0, solutions (14) and (15) to the Cauchy problem (12) and (13) satisfy the following estimates: and where S ≥ 1 is an integer, then the Cauchy problem admits the following estimate: where c 0 is a positive constant independent of t, k � (k 1 , k 2 , . . . , k n ) is a multi-index,|k| � k 1 + k 2 + · · · + k n , and D k x � z |k| /zx

Lemma 3.
Under assumption 1/r � 1/p + 1/q, if all norms appearing on the right-hand side below are bounded, then for any given integer S ≥ 0, we have then For any given integer and such that all norms appearing on the right-hand side below are bounded, then where C S is a positive constant (depending on υ 0 ) and In particular, we have where C S is a positive constant (depending on υ 0 ) and

Lemma 4. Suppose that F � F(w) satisfies the same assumptions as in Lemma 3. If vector functions w � w(x) and
w � w(x) satisfy (30), respectively, and such that all norms appearing on the right-hand side below are bounded, then for any given integer S ≥ 0, let We have Discrete Dynamics in Nature and Society where p, q and r satisfy 1/r � 1/p + 1/q and C S is a positive constant (depending on υ 0 ).

The Proof of Theorem 1
In this section, we shall discuss the Cauchy problem for n-dimensional nonlinear evolution equations: where For the nonlinear term f in (39), we give the following hypotheses: in a neighborhood of λ � 0, say, for |λ| ≤ 1, f � f(λ) is sufficiently smooth and where α is an integer ≥1. For any given integer S such that S ≥ n + 5, and any positive numbers E, T (0 < T ≤ + ∞), we introduce the following set of functions: where We now define a map where (u, v) are the solutions of the linear equations (47) e next step is to prove that the map M is a strict contraction in the complete metric space X S,E,T .

5.
For By (23), we have Using (29) and (33), we get A combination of (50)-(52) leads to Noting that S ≥ n + 5, by the definition of X S,E,T , we have On combining (53)-(57), we get We point out the fact that, if b ≥ a ≥ 0, then we have e proof of (59) was shown in [16], but to complete it, we include here. Let e combination of (60) and (61) gives (59). us, from (58) and (59), we obtain sup 0<t≤T ( By (49) and (24), we also have From (29) and (33), we have Moreover, in fact, us, we have Discrete Dynamics in Nature and Society 5 On combining (63)-(65) and (67) and the definition of X S,E,T , we obtain By (28), we have Noting the hypotheses of (43), it follows from (33) that Using (67), We can also get By (25), we have By (29) and (33), we have From (67), we have n deriving (76), we have used and 6 Discrete Dynamics in Nature and Society On combining (73)-(78), we obtain us, Finally, from (26) and (72), we have On combining (75) and (81), we obtain On combining the above discussions, we obtain Proof. By the definition of M, we have Discrete Dynamics in Nature and Society Noting that S ≥ n + 5, and the definition of X S,E,T and using the hypotheses (43) and (38) (in which we take r � 1, p � +∞, q � 1), we get For we have we also have us, we can get Putting (88) and (91) into (87), and using the definition of X S,E,T 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 ‖ · ‖ H S and the definition of D S,T (h, p), we get Moreover, we have us, we see Similarly, we have Besides, we have On combining (94)-(101), we obtain Noting that α is an integer ≥1, 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 p � +∞), we get Discrete Dynamics in Nature and Society 9 Still using (97), (99), and (101) and notifying that it follows that For we know that From (67) and (97), we obtain us, in deriving (111), we have used On combining (107), (111), and (112) and using Similar to (73), we obtain 10 Discrete Dynamics in Nature and Society We have from (38) that We can obtain and we also have For we have Using (99), we obtain Following the same procedure as (122), we obtain Discrete Dynamics in Nature and Society (1 + t) − n dτ .
We can easily get us, we have Now, we need to seek a new method to give a bound for So, we can easily see that On combining (125)-(132), we obtain ] Using (38), we obtain 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 η ⟶ 0, our method is no longer applicable. ere 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
e authors declare that they have no conflicts of interest.