The global Goursat problem on \(R\times S^ 1\)
The authors study regularity for the Goursat problem for nonlinear wave equation on \(R\times S^ 1\). In the Goursat problem, the data are given on a light cone and lacks the domain dependence properties associated with the solution of the Cauchy problem. Also, the infinite generator of the evolution is nonlocal. For a given arbitrary data in the Sobolev space H(C), it is shown under suitable assumptions that the Goursat problem has a unique solution locally in time. Necessary and sufficient conditions are obtained for the local solution to be \(C^ 2\). The authors have proved estimates from Goursat data to Goursat data, from Goursat data to Cauchy data, and from Cauchy data to Goursat data. Under appropriate conditions, a global solution in H(C) is obtained for the Goursat problem.
scientific article; zbMATH DE number 4152931