Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients (Q2415187)

There is an investigation on the Cauchy problem for general second order strictly hyperbolic equations in divergence form. Under some special assumptions for coefficients, parameters and initial data, the existence of a unique weak solution is proved. The lifespan of the solution is improved. The proofs follow just classical steps. After presenting basic elements including paradifferential calculus one rewrites the equation as a new system in \((u,v)\) and one obtains a-priori estimates for the smooth solution \((u,v)\). Next, the energy estimate for the weak solution of initial equation is proved followed by the complete proof of the local well-posedness.