Blow-up of solutions to a nonlinear wave equation

Georgiev, Svetlin Georgiev

Electronic journal of differential equations, Tome 2004 (2004)

Résumé

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].

Détail

Classification : 35L10, 35L50
Keywords: wave equation, blow-up, hyperbolic space