The Initial Value Problem For A Class Of Nonlinear Wave Equation With Damping Term

In this paper,we are concerned with the initial value problem for a class of N-Dimensional nonlinear wave equation with damping termutt-△utt-△u-△ut=f(u)-△g(u),x∈Rn,t>0,(1)u(x,0)=u0(x), ut(x,0)=u1(x), x∈Rn,(2) and the initial value problemutt-△utt-△u-△ut=△f(u),(3)u(x,0)=u0(x),ut(x,0)=u1(x).(4) where u(x,t) denotes the unknown function, f(s) is the given nonlinear function,u0(x) and u1(x) are given initial value functions, the subscript t indicates the partial derivative with respect to t,n is the dimension of space variable x, and△denotes the Laplace operator in Rn.This paper is divided into four chapters.The first chapter is the introduction.In the second chapter, we will study the existence and uniqueness of the local solution to the ini-tial value problem (1)(2) in W2’p(Rn)∩L∞(Rn),then we will give the extension theorem of the solution. Using integral estimation, we will prove the existence and uniqueness of the global solution to the initial value problem (1)(2) in W2>p(Rn)∩L∞(Rn).In the third chapter, we will study the existence and uniqueness of the local solution to the initial value problem (1)(2) in Hs(Rn),then we will give the extension theorem of the solution. Using integral estimation, we will prove the existence and uniqueness of the global solution to the initial value problem (1)(2) in Hs(Rn). In the fourth chapter,we will study the asymptotic behavior of solution of the global solution to the initial value problem(3)(4). The main results are the following:Theorem1Assume that u0,u1∈W2,p(Rn)∩L∞(Rn),f(s),g(s)∈C3(R) and f(0)=0,then problem (1)(2) has a unique local solution u(x,t) E C3([0,T0); W2,p(Rn) ∩L∞(Rn)),where[0,T0)is a maximal time interval.if then To=∞.Theorem2Assume that u0,u1∈W.,p(Rn)∩L∞(R"),f(s),g(s)∈C’3(R),f(0)=g(0)=0,(?)∈R,g’(s)≤A0,|f’(s)-g’(s)|≤A0,then problem(1)(2)has a unique global solution u(x,t)∈C3([0,∞]；W2,p(Rn)∩L∞(Rn)).Theorem3Assume that s>n/2and v0,v1∈Hs,f,g∈C[s]+1and f(0)-g(0)=0,then problem(1)(2) has a unique local solution v(x,t)∈C2([0,T0)；Hs),where[0,T0) is a maximal time interval.if then T0=∞.Theorem4Assume that v0,v1∈Hs(s>n/2),f,g∈C[s]+1(R),f(0)=g(0)-0and (?)s∈R,g’(s)≤A1,|f’(s)-g’(s)|≤A1,then problem(1)(2)has a unique global solution v(x,t)∈C’2([0,∞)；Hs).Theorem5Let q,γ,s be positive numbers such that q∈[1,2],γ≥0,n/q+n/2-1≤k, α≥2,Then there exists a positive constant δ,such that for any u0∈Hq-2γ,u1∈Hq-2γ-1satisfying problem(3)(4)has a unique global solution u∈C’([0,∞)；Hs).Moreover, where the small positive constant p depends only on f and δ.