Partial Differential Equations

Answer: The famous Hadamard example is: ∆u = 0 in ℝ², u(x,0) = exp(–k ^1/2)coskx, u _y (x,0) = 0 with the solution u(x,y) = exp(–k ^1/2)(coskx)(coshky). As k→∞, the Cauchy data and their derivatives of each order tend uniformly to zero, but the solutions are unbounded if y = 0. For example: [M] Ch. 1, Sec. 1, Ex. 3.