Initial boundary value problem for generalized Zakharov equations with nonlinear function terms

In this paper, we consider the initial boundary value problem for generalized Zakharov equations. Firstly, we prove the existence and uniqueness of the global smooth solution to the problem by a priori integral estimates, the Galerkin method, and compactness theory. Furthermore, we discuss the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero.

where the parameters β > 0 and α are real numbers. You and Ning [15] considered the existence and uniqueness of the global smooth solution for the initial value problem of the following generalized Zakharov equations in dimension two: where ε(x, t) = (ε 1 (x, t), ε 2 (x, t), . . . , ε N (x, t)) is an N -dimensional complex-valued unknown functional vector, v(x, t) = (v 1 (x, t), v 2 (x, t)) is a two-dimensional real-valued unknown functional vector, n(x, t) is a real-valued unknown function, x ∈ R 2 , and ϕ(s) is a real function.
In the present paper, we study the following initial boundary value problem for generalized Zakharov equations: with initial data ε| t=0 = ε 0 (x), v| t=0 = v 0 (x), n| t=0 = n 0 (x), x ∈ [0, L], (1.4) and boundary conditions ε(0, t) = ε(L, t) = v(0, t) = v(L, t) = n(0, t) = n(L, t) = 0, (1.5) where the parameters p > 0, β > 0, α, and δ are real numbers, and ϕ(s) is a real function. Taking δ = 0, β = 0, and ϕ (s) = Constant in this system, it becomes the classical Zakharov equation system. From a physical point of view, this system has stronger nonlinear excitation and interaction. It also can be considered as a further generalization of the generalized Zakharov system discussed in [14]. From the perspective of both mathematical research and physical applications, the problem is of great significance. For convenience of the following contexts, we set some notations. For 1 ≤ p ≤ ∞, we denote by L p [0, L] the space of all pth-power integrable functions in [0, L] equipped with norm · L p , and by H s,p the Sobolev space with norm · H s,p . For p = 2, we write H s instead of H s,2 . C k (R) is the space of k times continuously differentiable functions on R. If k = 0, then we write C(R) instead of C 0 (R). Let (f , g) = L 0 f (x)g(x) dx, where g(x) denotes the complex conjugate function of g(x). The real and imaginary parts of a complex number A are denoted, respectively, by Re A and Im A. Throughout the paper, C is a generic constant, which may have different meanings in different places.
This paper is organized as follows. In Sect. 2, we establish a priori estimations for problem (1.1)-(1.5). In Sect. 3, we study the existence and uniqueness of global generalized solutions for problem (1.1)-(1.5). In Sect. 4, we discuss the regularity of global generalized solution for problem (1.1)-(1.5). In Sect. 5, we give the approximation limit of the global solution when the coefficient of the strong nonlinear term tends to zero.
Proof Taking the inner product of (1.1) and ε, we have iε t + ε xx + (αn)ε + δ|ε| p ε, ε = 0. (2.1) Since Im(iε t , ε) = 1 2 d dt ε 2 L 2 , Im(ε xx + (αn)ε + δ|ε| p ε, ε) = 0, and hence from (2.1) we get d dt ε 2 L 2 = 0, that is, Then for the solution of problem (1.1)-(1.5), we have Proof Taking the inner product of (1.1) and -ε t , we get that we have Taking the inner product of (1.2) and v, we have and hence from (2.4) we get d dt By (2.3) and (2.5) we get and C is a positive constant depending only on n, m, j, q, r, and θ . (2) For γ > 0 and s ∈ Z + , we can get a constant C (it only depends on γ and s) such that Proof From Lemmas 2.1-2.3 and Young's inequality we get and hence Proof Differentiating (1.1) with respect to t, we get Taking the inner product of (2.7) and ε t , it follows that and hence from (2.8), (1.3), and Corollary 2.1 we get Taking the inner product of (1.2) and -v xx , it follows that and hence from (2.10) we get By (2.9) and (2.11) we obtain and thus by Gronwall's inequality we obtain The Lemma 2.5 is proved.
Proof By Lemmas 2.3 and 2.5 the result of Corollary 2.2 is obvious.

Lemma 2.6
Suppose that the conditions of Lemma 2.5 are satisfied, and assume that Proof Taking the inner product of (2.7) and -ε txx , it follows that (2.14) Since Taking the inner product of (1.2) and v xxxx , it follows that and thus by Gronwall's inequality we obtain By (1.1), Young's inequality, and (2.18) we obtain Proof By Lemmas 2.3 and 2.6 the result of Corollary 2.3 is obvious.
Proof We prove this lemma by mathematical induction. By Lemma 2.6 the lemma is true Next, we will show that the lemma is true for l = k + 1.

from (1.3) and (3.25) we get
(3.26) By (3.24) and (3.26) we obtain By using Gronwall's inequality we obtain and hence Therefore the proof of Theorem 3.2 is completed.

The regularity of global generalized solution for problem (1.1)-(1.5)
To get the regularity of the global generalized solution for problem (1.1)-(1.5), we need the following lemma and corollary.
By (1.2) and Lemma 2.6 we obtain Proof By Lemmas 2.3 and 4.1 the result of Corollary 4.1 is obvious.
Proof By using Theorem 3.1, Lemma 4.1, and Corollary 4.1 we can easily get this theorem. Proof By using Theorem 4.1 and the embedding theorems of Sobolev spaces we can easily get this theorem.
Proof By Lemma 2.7 and the embedding theorems of Sobolev spaces the result of Theorem 4.3 is obvious.

Approximation of solution
We now suppose that the generalized solution of initial boundary value problem (1.1)-(1.5) is approximated by the generalized solution of the following problem: with initial data and boundary conditions where the parameters p > 0, β > 0, and α are real numbers, and ϕ(s) is a real function. Letting with initial data Proof Taking the inner product of (5.6) and F, it follows that Taking the inner product of (5.7) and G, it follows that by (5.12), (5.14), and (5.15) we get Differentiating (5.6) with respect to t, we get Taking the inner product of (5.17) and G t , it follows that Taking the inner product of (5.7) and -G xx , it follows that from (5.8) and (5.20) we get Taking the inner product of (5.11) and F xx , it follows that By using Gronwall's inequality we obtain By (5.8) we get By (5.6) and (5.24) we obtain Proof Taking the inner product of (5.17) and -F txx , it follows that Im δ|ε| p ε t , -F txx = -Im Taking the inner product of (5.7) and G x 4 , it follows that (H x , G x 4 ) = -