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

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+) = ∞.