Global Regularity And Finite Time Blow-up Of Hyperbolic Equations

Solutions of hyperbolic equation has global existence, or blows up infinite time. Theories show that there exists a critical state, i.e., between thetwo sides, the nature of the critical state of solution is opposite. Forhyperbolic coupled systems, as for hyperbolic equation, it can obtain solutionproperties through the study of solution estimation, the key lies in how todeconstruct the equation, and a wide method is Riemann invariants innumerous ways.Through the existing results, combining the Riemann invariants, prioriestimate and undetermined coefficient method, the paper make research onthree models like P system, traffic flow model, Euler equation. Firstly, withthe help of Riemann invariants, the original system is diagonalized by alongthe characteristic line and calculated, then it can get a priori estimate of theRiemann invariants itself. Next, considering the derivative of Riemanninvariants as new dependent variables, rebuild a new equation, and then combine with the method of undetermined coefficients and the originalequation, the new system will be deconstructed or part of deconstruction,bringing it into ordinary differential equations along the characteristic linemethod, in order to estimate Riemann invariant derivative, finally combinetwo parts, give the upper limit on global regularity and lower limit on finitetime blow up of the three model solution in the different critical conditions.The conclusion indicates that the next work and research directions.