Uniform continuity of the solution map for nonlinear wave equation in Reissner-Nordstrom metric

S. Georgiev, University of Sofia, Sofia, Bulgaria

E. J. Qualitative Theory of Diff. Equ., No. 12. (2007), pp. 1-14.

Abstract: In this paper we study the properties of the solutions to the Cauchy problem
$$
(u_{tt}-\Delta u)_{g_s}=f(u)+g(|x|),\quad t\in [0, 1], x\in {\cal R}^3,
\leqno{(1)}
$$
$$
u(1, x)=u_0\in {\dot H}^1({\cal R}^3),\quad
u_t(1, x)=u_1\in L^2({\cal R}^3),
\leqno{(2)}
$$
where $g_s$ is the Reissner-Nordstr${\ddot o}$m metric (see [2]); $f\in {\cal C}^1({\cal R}^1)$, $f(0)=0$, $a|u|\leq f'(u)\leq b|u|$, $g\in {\cal C}({\cal R}^+)$, $g(|x|)\geq 0$, $g(|x|)=0$ for $|x|\geq r_1$, $a$ and $b$ are positive constants, $r_1>0$ is suitable chosen. When $g(r)\equiv 0$ we prove that the Cauchy problem $(1)$, $(2)$ has a nontrivial solution $u(t, r)$ in the form $u(t, r)=v(t)\omega(r)\in {\cal C}((0, 1]{\dot H}^1({\cal R}^+))$, where $r=|x|$, and the solution map is not uniformly continuous. When $g(r)\ne 0$ we prove that the Cauchy problem $(1)$, $(2)$ has a nontrivial solution $u(t, r)$ in the form $u(t, r)=v(t)\omega(r)\in {\cal C}((0, 1]{\dot H}^1({\cal R}^+))$, where $r=|x|$, and the solution map is not uniformly continuous.

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