The Cauchy Problem For A N-dimensional Generalized Modified Benney-luke Equation

This paper consists of four chapters. The first chapter is the introduction. In thesecond chapter, we will study the existence and uniqueness of the local solution for theCauchy problem of a n-dimensional generalized modified Benney-Luke equation. In thethird chapter, we will study the existence and uniqueness of the global solution to theCauchy problem for the above mentioned equation for theβ≤0 case. In the fourthchapter, we will discuss global existence and blow-up of the solution to the Cauchyproblem for the above mentioned equation for theβ＞0 case . The details are these:In the second chapter ,we study the following Cauchy problem for a n-dimensionalgeneralized modified Benney-Luke equation:whereμ,ε,α,β,a,b,A,B and p are positive real constants. u(x, t) denotes the unknownfunction, subscripts t and x indicate the partial derivative with respect to t and x, andp satisfies the following condition:For this purpose, we will get the following equivalent form of the problem (1):where Then, using the contraction mapping principle ,we can prove the existence anduniqueness of the local solution for the Cauchy problem (1), (2). The main result is thefollowing:Theorem 1 Suppose thatφ∈H³,ψ∈H², then the Cauchy problem (1), (2) hasa unique local solution u∈C([0,T];H³)∩C¹([0,T];H²),(?)T∈(0,T₀), where [0,T₀) isthe maximal time interval of existence of u(x, t). Moreover, ifthen T₀=∞.In Chapter 3, we give the energy conservation law and prove the existence anduniqueness of the global solution to the Cauchy problem (1), (2) for theβ≤0 case. Themain results are the following:Theorem 2 Suppose thatφ∈H³,ψ∈H²,β≤0, then the Cauchy problem(1), (2) has a unique global solution u∈C([0,∞);H³)∩C¹([0,∞);H²)∩C²([0,∞);H¹).In Chapter 4, we will discuss global existence and blow-up of the solution to theCauchy problem (1),(2) for theβ＞0 case. LetDefine the stable set (potential well) and unstable set, respectively, byThe main result is the following: Theorem 3 Suppose thatφ∈H³,ψ∈H²,β＞0,E(0)＜d,I(φ)＞0or(?)=0,then the Cauchy problem (1),(2) has a unique global solution u∈C([0,∞);H³)∩C¹([0,∞);H²)∩C²([0,∞);H¹).Theorem 4 Suppose thatα=0,β＞0,φ∈H³,ψ∈H²,E(0)＜d and I(φ)＜0.If u(x,t) is a local solution to the Cauchy problem (1), (2) on [0,T₀), then T₀＜+∞.