Theory of Partial Differential Equations (Enhanced Edition‪)‬

Let us consider under what conditions it is possible to determine a unique solution of the equation u,, - uyy = 0 (1.1.1) satisfying the conditions u(x,O) = f(x) and uy(x,O) = g(x) (1.1.2) where f: (a,b)+R' and g: (a,b)+ R'. We understand that as part of this
task we are to decide precisely what we wish to mean by saying a function u
is a solution of (l.l.l), (1.1.2) and what properties given functions f and g
must have so that such a solution exists and is unique.