Blow-up of solutions to a nonlinear wave equation

Abstract

We study the solutions to the radial 2-dimensional wave equation Xtt - 1/ r Xr - Xrr + sinh2X/ 2r2 = g, X(1, r) = X∘ ∈ Ḣγ rad, Xt(1, r) = X1 ∈ Ḣγ-1rad, where r = |x| and x in ℝ2. 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 Lp ([0, 1]Lqrad), with 1/p + 2/q = 3 - γ, 0 < γ < 1, p ≥ 1, and 1 < q ≤ 2, for which the solution satisfies limt→0 ∥x̄∥Ḣγ rad = ∞. In doing so, we provide a counterexample to estimates in [1].