Electron. J. Diff. Eqns., Vol. 2004(2004), No. 77, pp. 1-7.

Blow-up of solutions to a nonlinear wave equation

Svetlin G. Georgiev

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 ${1\over p}+{2\over q}=3-\gamma$, $0 less than\gamma less than 1$, $p\geq 1$, and $1 less than q\leq 2$, 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.

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Svetlin Georgiev Georgiev
University of Sofia
Faculty of Mathematics and Informatics
Department of Differential Equations, Bulgaria
email: sgg2000bg@yahoo.com

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