Global Existence of Solution for Cauchy Problem of Two-Dimensional Boussinesq-Type Equation

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