Electron. J. Diff. Eqns., Vol. 2005(2005), No. 67, pp. 1-22.

Blow up of solutions for Klein-Gordon equations in the Reissner-Nordstrom metric

Svetlin G. Georgiev

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 $\mathcal{C}((0, 1]{\dot B}^{\gamma}_{p, p}(\mathbb{R}^+))$,
$$ u(t,r)= \cases v(t)\omega(r)&\hbox{for }t\in (0, 1],\; r\leq r_1\\ 0&\hbox{for } t\in (0, 1],\; r\geq r_1, \endcases $$
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.

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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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