Blow up of solutions for Klein-Gordon equations in the Reissner-Nordstrom metric
Date
2005-06-27
Authors
Georgiev, Svetlin G.
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
Abstract
In this paper, we study the solutions to the Cauchy problem
(utt - Δu)gs + m2</sup>u = ƒ(u), t ∈ (0, 1], x ∈ ℝ3,
u(1, x) = u0 ∈ Ḃγp,p (ℝ3), ut (1, x) = u1 ∈ Ḃγ-1p,p (ℝ3),
where gs is the Reissner-Nordströ m metric; p > 1, γ ∈ (0, 1), m ≠ 0 are constants, ƒ ∈ C2 (ℝ1), ƒ(0) = 0, 2m2|u| ≤ ƒ(l) (u) ≤ 3m2|u|, l = 0, 1. More precisely we prove that the Cauchy problem has unique nontrivial solution in C((0, 1] Ḃγp,p (ℝ+)),
u(t, r) = {v(t)ω(r) /0 for t ∈ (0, 1], r ≤ r1 for t ∈ (0, 1], r ≥ r1,
where r = |x|, and limt→0
u
Ḃγ p,p (ℝ+) = ∞.
u
Ḃγ p,p (ℝ+) = ∞.
Description
Keywords
Partial differential equation, Klein-Gordon, Blow up
Citation
Georgiev, S. G. (2005). Blow up of solutions for Klein-Gordon equations in the Reissner-Nordstrom metric. <i>Electronic Journal of Differential Equations, 2005</i>(67), pp. 1-22.
Rights
Attribution 4.0 International