Abstract

In this paper, we consider the Cauchy problem of two-dimensional Boussinesq-type equations . Under the assumptions that is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence of global weak solution.

1. Introduction

In this paper, we study the following Cauchy problem of two-dimensional Boussinesq-type equations: where denotes the unknown function, is the given nonlinear function with exponential growth like at the infinity, and are the given initial value functions, the subscript indicates the partial derivative with respect to , and denotes the Laplace operator in .

This model arises in a number of mathematical models of physical processes, for example, in the modeling of surface waves in shallow waters or in considering the physical study of nonlinear wave propagation in waveguide [1–3]. In the one dimensional case, the longitudinal displacement of the rod satisfies the following double dispersion equation (DDE) [1–3]: and the general cubic DDE (CDDE), where , and are positive constants. A great deal of efforts has been made to establish the sufficient condition for the existence or nonexistence of global solution to various nonlinear terms, such as or , or (see, e.g., [4–10]). In [7, 8] Chen et al. studied the initial boundary value problem and the Cauchy problem of the following generalized double dispersion equation which includes (4) as special cases: For the case (bounded below) they proved the existence of global solutions. And they also show the nonexistence of the global solution under some other conditions to deal with the global well posedness of (4). Recently, in [9, 10], for the nonlinear term satisfying more general conditions than both the convex function and , the Cauchy problem and the initial boundary value problem for (5) with , were studied, respectively. For both of above problems, the authors obtained the invariant sets and sharp conditions of global existence of solution by introducing a family of potential wells.

But to the authors' best knowledge, there are very few works on the multidimensional cases for the Cauchy problems (1) and (2). Most recently, in [11–17], the authors considered the Cauchy problem of the multidimensional nonlinear evolution equation for constant . Consider They gave the existence of local and global solution and the nonexistence of global solution. Note that, in [11–17], in order to obtain the global existence and the asymptotic of solution for problem (6), the authors requested that possesses polynomial growth. In [14], the authors requested or (bounded below), so the global existence of solution for Cauchy problem of (6) was solved. However, the results of [11–17] are not applicable for (1) with exponential growth like at the infinity. We point that the conditions of Lemma 2.2 and Corollary 2.3 in [14] are essentially power-type nonlinearity; that is, , where and are positive constants, although they requested or (bounded below). In [11, 12], Polat et al. studied the existence, both locally and globally in time, asymptotic, and blow up of a solution for the problem (6) under the assumption and .

In this paper, we consider the Cauchy problems (1) and (2) with nonlinear term like exponential growth at the infinity for two-dimensional case. The motivation of studying problems (1) and (2) with exponential growth nonlinear term comes from the following nonlinear damped wave equation: where is a bounded domain of . In the recent works of Ma and Soriano [18] and Alves and Cavalcanti [19], the authors assume that admits an exponential growth and show global existence as well as blow up of solutions for problem (7) with initial-boundary conditions. The ingredients used in their proof [18, 19] are essentially the Trudinger-Moser inequality (see [20, 21]) and the well-known Mountain Pass Level by Ambrosetti and Rabinowitz [22].

The main purpose of this paper is to establish the sufficient conditions for existence of global solution to the Cauchy problems (1) and (2) with exponential nonlinearities for spaces dimensions , by considering similar arguments due to [18, 19]. The ingredients used in our proof are essentially the Trudinger-Moser inequality when (see [23, 24]) and Galerkin methods. As far as we are concerned, this is the first work in the literature that take into account the exponential growth of the function for the Cauchy problems (1) and (2).

This paper is organized as follows. Section 2 is concerned with some notations and statement of assumptions. In Section 3, we prove the existence of global solutions to the Cauchy problems (1) and (2) by using Galerkin approximation scheme.

2. Preliminaries

In this paper, we denote by by , , , and . We consider problems (1) and (2) with initial data , , and . Here, , and , are the Fourier transformation and the inverse Fourier transformation, respectively. We also define the space with the norm and the space with the norm Since is dense in , we have that is dense in .

We call a weak solution of problems (1) and (2) on , if , , satisfying in , and

Throughout this paper we will make repeated use of the Trudinger-Moser inequality which can be found in [23, 24].

Proposition 1 (see [23, 24] (Trudinger-Moser inequality)). If and , then Moreover, if , then there exists a constant , such that
The nonlinear function is required to satisfy the following conditions:(H) is a continuous function; for each , there exists a positive constant such that and, for each , there exists a positive constant , which depends on  , verifyingfor some , where .
As in [15], we introduce two functionals as follows:

According to the Trudinger-Moser inequality (Proposition 1), if satisfies (H), then and are well defined on .

In order to prove our main results we also need the following inequality, for all and ,

3. Existence of Global Solution

In this section, we study the existence of global solution for problem (1)-(2). To this end, we begin this section by the following lemma.

Lemma 2. Suppose that satisfies the assumption (H) and assume that is a bounded sequence in . Then, if for some , there exists a subsequence of , which satisfies the following convergence:

Proof. Since is bounded, let such that for all and . Then selecting such that , we have from (17) and (21) that Using Trudinger-Moser inequality (15), we obtain Then, by (25), (26), and embedding theorem, we get for all . This shows that is a bounded sequence in . Besides, since is compact, there exists a subsequence of , still denoted by , such that a.e. in ; it follows from the continuity of that a.e. in . Then (23) follows from a well-known result by Lions ([25], Ch. 1, Lemma 1.3).
As the above arguing, using (18), the Trudinger-Moser inequalities (15) and (21), we obtain for all . Then, (24) holds.
Now, we consider the existence of global solution for problems (1) and (2).
Let , and be basis function systems in . Construct the approximate solutions of problems (1) and (2). Consider satisfying By multiplying (30) by and summing for , we get where

Lemma 3. Suppose that satisfies the assumption (H), , then and as . Furthermore, there exists positive constant which is independent of such that

Proof. Since (18), (14), and , we have then we get . It follows from (31) and (24) in Lemma 2 that From (31) and (32), we get that, as , By combining (38) with (39), we have as and then (36) is obvious.

Theorem 4. Let satisfy the assumption (H), . Then, for any , the problems (1) and (2) admit a global weak solution with .

Proof. For problems (1) and (2), construct the approximate solutions (30) and (31). From (34) and (36), we get
From (40) and (19), it follows that in and in are bounded, respectively. Hence, by Lemma 2, there exists a and a subsequence of , still denoted by , such that   in weakly star and a.e. in ;  in weakly star;  in weakly star.By integrating (40) with respect to from 0 to , we get Let in (41), we obtain
On the other hand, from (31) and (32) we have in and in . Therefore, the above is a global weak solution of problems (1) and (2).

4. Example and Generalization

In this section we give some examples of the conditions in Section 2 and some generalizations for the results obtained in Section 3.

A typical example of satisfying (H) is , for some , . Obviously, , and direct computations show that, for any , and , since , then (16) and (17) hold. From (17), we get (18). For , we have, for , Hence, (19) holds.

Another example of satisfying (H) is , for some . Obviously (16), (17), and (18) are satisfied.

We also can take , for some and is a continuous function with for constants and .

Consider the following two-dimensional double dispersion equation with damped: then (34) can be written as we can get the following result.

Theorem 5. Let satisfy the assumption (H), . Then, for any , the problems (44) and (2) admit a global weak solution with .

We can also consider the problems (44) and (2) in for and get the similar results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no.11171311) and the Natural Science Foundation of Henan Province (1323004100360).