The Cauchy Problem Of A Four-ordered Nonlinear Wave Equation

In this paper, we are concerned with the following Cauchy problem:where x∈R,t∈R⁺,β＞0 is a constant, u(x, t) denotes an unknown function, P,Φ,Ψare given nonlinear functions, u₀(x) and u₁(x) are given initial value functions, andsubscripts x and t indicate the partial derivative with respect to x and t, respectively.Here, for convenience, we let P'(0) = 0,Ψ(0) = 0.This paper consists of four chapters: The first chapter is the introduction; in the secondchapter, we will use contraction mapping priciple to study the existence and uniquenessof the local solution of the Cauchy problem(1) (2); in the third chapter, we will study theexistence and uniqueness of the globle solution of the Cauchy problem(1) (2) by meansof energy method and a priori estimate; in the fourth chapter, we will use the concavitymethod to study the nonexistence of the globle solution of the Cauchy problem(1) (2).The details are these:In the chapter 2, using the contraction mapping priciple, we study the existence anduniqueness of the local solution of the Cauchy problem(1) (2). The main results arefollowing:Theorem 1 Assume that u₀∈H¹∩W^1,∞,u₁∈H¹∩W^1,∞,|Φ'''(u)-Φ'''(v)|≤max{C₂,C₃|u-v|(1+|u|^m₁+|v|^m₁)},Φ'(0)=0,|P'(u)-P'(v)|≤C₄|uv|(1+|u|^m₂+|v|^m₂),|Ψ'(u)-Ψ'(v)|≤C₅|u-v|(1+|u|^m₃+|v|^m₃),Ψ'(0)=0, (3) where C_i,m_j≥0(i=2,3,4,5;j=1,2,3) , then there exists a maximal time T₀ whichdepends only on ||u₀||_H¹+||u_0x||_∞+||u₁||_H¹+||u_1x||_∞and the problem (1) (2) has a uniquesolution u withMoreover ifthen T₀=∞.Remark 1: IfΨ=Φ, we only delete (3), theorem 1 holds true.Theorem 2 Suppose that the assumptions of Theorem 1 hold, then there existsa maximal time T₀ which depends only on ||u₀||_H¹+||u_0x||_∞+||u₁||_H¹+||u_1x||_∞and theCauchy problem (1) (2) has a unique solution u withMoreover if (4) holds,then T₀ =∞.In the chapter 3, using energy method and a priori estimate, we prove the existenceand uniqueness of the global solution of the Cauchy problem(1) (2). The main results arefollowing:Theorem 3 Suppose that the assumptions of Theorem 1 hold, C₉ is a constant, andthe Cauchy problem (1) (2) has a unique global solution u(x, t) withRemark 2: IfΨ=Φ, we only change (5) to C₉≥0, theorem 3 holds true.Theorem 4 Suppose that the assumptions of Theorem 1 hold and (?), satisfying the Cauchy problem (1) (2) has a unique global solution u(x, t) withIn the chapter 4, the nonexistence of the globle solution of the Cauchy problem(1) (2)is gived by means of the concavity method. The main results are following:Theorem 5 Suppose that the assumptions of Theorem 1 hold, there exists a constantα＞0 such thatand one of the following two assumptions holds:(1)E(0)＜0(2)E(0)≥0,(u₁+u₀)+β(u_1x,u₁)＞(?)Then the solution u(x,t) of the problem (1) (2) cease to exit in finite time.Remark 3 IfΨ=Φ, delete (6) , theorem 5 holds true.