The Cauchy Problem For A Class Of Nonlinear Wave Equations With Damping Term

In the paper, we study the following Cauchy problem for a class of nonlinear waveequations with damping term:whereα＞0,b＞0 are constants, u(x, t) denots the unknown functions, subcripts x andt indicate partial derivatives, f(s) is given nonlinear function, u₀(x) and u₁(x) are giveninitial value functions in R.This paper consists of five chapters: The first chapter is the introduction. In thesecond chapter, we will study periodic boundary value problem of the equation (1). In thethird chapter, we will study the existence and uniqueness of the global generalized solutionand classical solution for the Cauchy problem (1)-(2). In the fourth chapter, we will usethe cancavity method to study the blow up of the solution to the problem (1)-(2) and wewill take an example to illustrate that there exist the functions satisfied conditions of blowup of the solution. The main results are the following:Theorem 1 Suppose that u₀∈H⁴(Ω) and u₁∈H²(Ω) are periodic functions of xwith period 2D, f(s)∈C²(R) and f'(s) is bounded blow, i.e. there is a constant C₀, suchthat f'(s)≥C₀, (?)∈R. Then the periodic boundary value problem of the equation (1)has a unique global generalized solution whereΩ=(-D,D) and u(x, t) satisfies the periodic boundary condiation (3) and initialcondiation (4) in the classical meaning.Theorem 2 Suppose that u₀∈H⁷(Ω) and u₁∈H⁵(Ω) are periodic functions of xwith period 2D, f(s)∈C⁵(R) and f'(s) is bounded blow. Then the periodic boundaryvalue problem (1), (3), (4) has a unique global classical solutionTheorem 3 Suppose that u₀∈H⁴(R),u₁∈H²(R),f(s)∈C²(R) and f'(s) isbounded blow. Then Cauchy problem (1), (2) has a unique global generalized solutionwhere u(x, t) satisfies initial condiation in the classical meaning.If we suppose that u₀∈H⁷(R),u₁∈H⁵(R),f(s)∈C⁵(R) and f'(s) is bounded blow.Then Cauchy problem (1), (2) has a unique global classical solutionTheorem 4 Suppose thatα＞0,b＞0,f(s)∈C(R),u₀∈H²(R),u₁∈L²(R), F(s) =(?),F(u_0x)∈L¹(R)and there existsβ＞0,such thatThen the generalized solution u(x, t) or the classical solution u(x, t) of Cauchy problem(1), (2) blows up in finite time if one of the following conditions holds:(1)E(0)＜0;(2)E(0)=0 andβ(u₀,u₁)＞b||u_0x||²;where E(t) =E(t)=||u_t||²+α||u_xx||²+2(?)+4b(?) and ||·||=||·||_L²(R).