Blow Up Of Solutions For Coupled Nonlinear Wave Equations

This thesis mainly studies two problems,one of which is studying lower bound for blow up time of solutions for coupled nonlinear wave equations,and the other is studying blow up of the solution for a class of couple nonlinear viscoelastic wave equations.The methods employed are mainly constructing corresponding auxiliary function and some important methods about the theory of partial differential equations.The thesis includes three chapters.In chapter 1,we provide a simple research summary of blow up problems for coupled nonlinear wave equations.In chapter 2,the estimation of lower bounds for blow up time of solutions for the following coupled nonlinear wave equation are considered.utt-Δu+ut= f1(u,v),(x,t)∈×∈Ω(0,T),vtt-Δv + vt =f2(u,v),(x,t)∈×∈Ω(0,T),u(x,t)= v(x,t)= 0,(x,t)∈(?)Ω ×(0,T),u(x,0)= u0(x),ut(x,0)= u1(x),x ∈ Ω,v(x,0)= v0(x),vt(x,0)= v1(x),x∈Ω,where Q is a bounded domain with smooth boundary(?)Ω in Rn,f(·,·):R2→R,i = 1,2 f1(u,v)and f2(u,v)satisfy the following equations f1(u,v)= a|u + v|2(r+1)(u + v)+ b|u|ru|v|r+2,f2(u,v)= a|u + v|2(r+1)(u + v)+ b|v|r v|u|r+2,where a,b,r are constants,and a,b>0,r satisfies-1<r,n = 1,2,1-<r<0,n≥3The system energy was defined and corresponding auxiliary function was constructed.An inequality about satisfying auxiliary function was obtained by estimating the system energy and the estimation of lower bounds for blow-up time of solution was proved.In chapter 3,we study the following wave equations,utt-Δu +∫0tg(t-s)Δu(x,s)ds + |ut|m-1ut(x,t)∈Ω×(0,T),-Δut +Δutt=f1(u,v),vtt-Δv+∫0th(t-s)Δv(x,s)ds+|vt|r-ivt(x,t)∈Ω×(0,T),-Δvt-vtt=f2(u,v),u(x,t)= 0,v(x,t)= 0,(x,t)∈(?)Ω ×(0,T),u(x,0)= u0(x),ut(x,0)=u1(x)x∈Ω,v(x,0)= v0(x),vt(x,0)= v1(x),x∈Ω,where Ω is a bounded domain with smooth boundary(?)Ω in Rn,g and h are given functions,u0(x),u1(x),v0(x)and v1(x)are initial value functions,f1(u,v)and f2(u,v)satisfy the following equations fx(u,v)= a|u + v|2(ρ+1)(u + v)+ b|u|ρ|v|(ρ+2),f2(u,v)= a|u + v|2(ρ+1)(u + v)+ b|v|ρ v|u|(ρ+2),where a,b>0,p satisfies-1<ρ,n= 1,2,-1<ρ≤3-n/n-2,n≥3.By constructing the auxiliary function,using some inequations and estimating the corresponding auxiliary function,this paper proves the equation blows up in a finite time and obtains the upper bound for blow-up time of solution.