Qualitative Analysis Of The Solution To Hyperbolic Equation

We are concerned with the properties of the solutions to single conservation laws and systems of one dimensional zero-pressure fluid. Using a potential function, we consider the relationship between its minimizing value points and characteristic. Finally, we divide all characteristics into two kinds and give out the specific classification criteria and judgments based on properties the characteristics.Futher more, we are concerned with the asymptotic behavior of solution to quasilinear hyperbolic equation for conversation laws. If there exists a unique nondegenerate minimizing value point, the minimizing function is smoth on its neighborhood; If there exist two nondegenerate minimizing value points, the shock wave divides the neighborhood into two components, the minimizing function is smoth on both components and discontinuous on the shock wave. Through a potential function and considering its minimizing value points, we show that the solution is piecewise smooth, and then consider the characteristics and the properties of the potential in different regions. At last, based on Schaeffer’s result on asymptotic analysis we can obtain the asymptotic behavior more precisely and qualitatively.