1.4Well Posed ProblemsThe set of functions used to formulate a PDE, which might include coefficients orterms in the equation itself as well as boundary and initial conditions, is collectivelyreferred to as the inputdata. The most basic question for any PDE is whether asolution exists for a given set of data. However, for most purposes we want to requiresomething more. A PDE problem is said to bewell posedif, for a given set of data:1. A solution exists.2. The solution is uniquely determined by the data.3. The solution depends continuously on the data.These criteria were formulated by Jacques Hadamard in 1902. The first two propertieshold for ODE under rather general assumptions, but not necessarily for PDE. It iseasy to find nonlinear equations that admit no solutions, and even in the linear casethere is no guarantee.The third condition, continuous dependence on the input data, is sometimes calledstability. One practical justification for this requirement is it is not possible to specifyinput data with absolute accuracy. Stability implies that the effects of small variationsin the data can be controlled.For certain PDE, especially the classical linear cases, we have a good under-standing of the requirements for well-posedness. For other important problems, for