Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 77, pp. 1-7.

Title: Blow-up of solutions to a nonlinear wave equation

Author: Svetlin G. Georgiev (Univ. of Sofia, Bulgaria)
 
Abstract:
 We study the solutions to the the radial 2-dimensional wave equation
 $$\displaylines{
 \chi_{tt}-{1\over r}\chi_r-\chi_{rr}+{{\sinh2\chi}\over {2r^2}}=g, \cr
 \chi(1, r)=\chi_{\circ}\in {\dot H}^{\gamma}_{\rm rad},\quad
 \chi_t(1, r)=\chi_1 \in {\dot H}^{\gamma-1}_{\rm rad},
 }$$
 where $r=|x|$ and $x$ in $\mathbb{R}^2$.
 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 $L^p([0,1]L_{\rm rad}^q)$,
 with
 ${1\over p}+{2\over q}=3-\gamma$, $0<\gamma<1$, $p\geq 1$, and
 $1<q\leq 2$, for which the solution satisfies
 $\lim_{t\to 0}\|{\bar \chi}\|_{{\dot H}^{\gamma}_{\rm rad}}=\infty$.
 In doing so, we provide a counterexample to estimates
 in [1].
  
Submitted March 16, 2004. Published May 26, 2004.
Math Subject Classifications: 35L10, 35L50
Key Words: Wave equation; blow-up; hyperbolic space.