Research On The Global Solution And Blow-up Properties Of One-dimensional Quasi-linear Wave Equation

In this paper,we are concerned with the Cauchy problem of the quasi-linear wave equation as follows in above equations,u(t,x)is an unknown real valued function and c∈C∞((θ0,∞)),θ0∈(-∞,0),a>0,A ∈R.In this paper,considering the global existence of the solution,we mainly considered the degenerative equation and the geometrical blow-up properties of the solution in this problem(1).We assumed u1(x)±c(u0(x))a(?)xu0(x)≤0,when A>0,-∫Ru1(x)dx<∫θ00 c(θ)adθ,the equation(1)exists the global solution;when A=0,-∫Ru1(x)dx>2 ∫θ00c(θ)adθ,the equation(1)generates the degeneration using characteristics and cut-off functions.We assumed u1(x)± c(u0(x))a(?)xu0(x)>0,when the compactness of initial values,the equation(1)generates geometrical blow-up phenomenon.