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.

Show me the PDF file (288K), TEX file, and other files for this article.

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

Return to the EJDE web page