Blowup for Nonlinear Hyperbolic Equations

Abstract

Part 1 The two basic blowup mechanisms: the ODE mechanism - systms of ODE, strictly hyperbolic semilinear systems in the plane, semilinear wave equations the geometric blowup mechanism - Burgers' equation and the method of characteristics, blowup of a quasilinear system, blowup solutions, how to solve the blowup system, how ...u blows up, singular solutions and explosive solutions combinations of the two mechanisms - which mechanism takes place first?, simultaneous occurrence of the two mechanisms. Part 2 First concepts on global Cauchy problems: short time existence lifespan and blowup criterion blow-up or not? functional methods Burgers' equation semilinear wave equation the Euler system blowup or not? comparison and averaging methods. Part 3 Semilinear wave equations: semilinear blowup criterion maximal influence domain maximal influence domain for weak solutions blowup rates at the boundary of the maximal influence domain an example of a sharp estimate of the lifespan. Part 4 Quasilinear equations in one space dimension: the scalar case Riemann invariants, simple waves and L1-boundedness the case of 2x2 systems general systems with small data rotationally invariant wave equations. Part 5 Nonlinear geometrical optics and applications: quasilinear systems in one space dimension - formal analysis, slow time and reduced equations, existence, approximation and blowup quasilinear wave equations - formal analysis, slow time and reduced equations, existence, null conditions, blowup further results on the wave equation - formal analysis near the boundary of the light cone, slow time and reduced equations, a local blowup problem, asymptotic lifespan for the two-dimensional wave equation.