FORMATION OF SINGULARITIES FOR VISCOSITY SOLUTIONS OF HAMILTON-JACOBIEQUATIONS IN ONE SPACE VARIABLE

In this work we study the generation and propagation of singularities (shock waves) of the solution of the Cauchy problem for Hamilton-Jacob i equations in one space variable, under no assumption on the convexit y or concavity of the hamiltonian. We study the problem in the class o f viscosity solutions, which is the correct class of weak solutions. W e obtain the exact global structure of the shock waves by studying the way the characteristics cross. We construct the viscosity solution by either selecting a single-valued branch of the multi-valued function given as a solution by the method of characteristics or constructing e xplicitly the proper rarefaction waves.