BLOW-UP OF SOLUTIONS TO A NONLINEAR WAVE EQUATION

We study the solutions to the the radial 2-dimensional wave equation [GRAPHICS] where r = vertical bar x vertical bar and x in R-2. We show that this Cauchy problem, with values into a hyperbolic space, is ill posed in subcritical Sobolev spaces. In particular, we construct a function g(t, r) in the space L-p([ 0, 1] L-rad(q)), with 1/p + 2/q = 3-gamma, 0 < gamma < 1, p >= 1, and 1 < q <= 2, for which the solution satisfies lim(t) (-> 0) parallel to(chi) over bar parallel to(H) over dot(rad)(gamma) = infinity. In doing so, we provide a counterexample to estimates in [1].