Partial Differential Equations That Are Hard to Classify

Abstract

Semi-linear n×n systems of the form A∂u/∂x+B∂u/∂y=f can generally be solved, at least locally, provided data are imposed on non-characteristic curves. There are at most n characteristic curves and they are determined by the coefficient matrices on the left-hand sides of the equations. We consider cases where such problems become degenerate as a result of ambiguity associated with the definition of characteristic curves. In such cases, the existence of solutions requires restrictions on the data and solutions might not be unique.