8.10 (Laplace's Equation Should Not Be Viewed as an Evolution Equation) Sup- pose we consider Laplace's equation in dimension N = 2 and think of y as time t. Thus, we consider the initial value problem un + uxx = 0 with u(x, 0) = f(x) and u,(x,0) = g(x). Check directly (i.e., differentiate) that for any positive integer n, if f(x) = 0 and gn(x) = sin nx , then a solution is un(x, () = sinh nt sin nx 12 As n tends to infinity, the initial data tends to zero (in fact uniformly). But what happens to the solution at any fixed later time ? Hence, is the problem well-posed in terms of stability with respect to small changes in the initial data