Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 67, pp. 1-22.

Title: Blow up of solutions for Klein-Gordon equations in
       the Reissner-Nordstrom metric
       
Author: Svetlin G. Georgiev (Univ.of Sofia, Bulgaria)

Abstract: 
 In this paper, we study  the solutions to the
 Cauchy problem
 $$\displaylines{
 (u_{tt}-\Delta u)_{g_s}+m^2u=f(u),\quad t\in (0, 1], x\in \mathbb{R}^3,\cr
 u(1, x)=u_0\in {\dot B}^{\gamma}_{p, p}(\mathbb{R}^3),\quad
 u_t(1, x)=u_1\in {\dot B}^{\gamma-1}_{p, p}(\mathbb{R}^3),
 }$$
 where $g_s$ is the Reissner-Nordstrom metric; $p>1$,
 $\gamma\in (0, 1)$, $m\ne 0$ are constants,
 $f\in \mathcal{C}^2(\mathbb{R}^1)$, $f(0)=0$,
 $2m^2 |u|\leq f^{(l)}(u)\leq 3m^2 |u|$, $l=0, 1$. More precisely we prove that
 the Cauchy problem has unique nontrivial solution in
 $\mathcal{C}((0, 1]{\dot B}^{\gamma}_{p, p}(\mathbb{R}^+))$,
 $$
 u(t,r)= \begin{cases}
 v(t)\omega(r)&\hbox{for }t\in (0, 1],\; r\leq r_1\\
 0&\hbox{for } t\in (0, 1],\; r\geq r_1,
 \end{cases}
 $$
 where $r=|x|$, and
 $\lim_{t\to 0}\|u\|_{{\dot B}^{\gamma}_{p, p}(\mathbb{R}^+)}=\infty$.
  
Submitted March 14, 2005. Published June 27, 2005.
Math Subject Classifications: 35L05, 35L15.
Key Words: Partial differential equation; Klein-Gordon; blow up.