Abstract
We construct for every fixed the metric , where , , , , are continuous functions, , for which we consider the Cauchy problem , where , ; , , where , , , , , , and are positive constants. When , we prove that the above Cauchy problem has a nontrivial solution in the form for which . When , we prove that the above Cauchy problem has a nontrivial solution in the form for which .
Highlights
In this paper, we study the properties of the solutions of the Cauchy problem utt − Δu gs = f (u) + g |x|, x ∈ Rn, n ≥ 2, (1)u(1, x) = u◦(x) ∈ L2 Rn, ut(1, x) = u1(x) ∈ H −1 Rn, (2)where gs is the metric gs = h1(r)dt2 − h2(r)dr2 − k1(ω)dω12 − · · · − kn−1(ω)dωn2−1, (1.1) ∞ 0h2(s) ∞ h1(s) s h2(τ) h1(τ) dτ < ∞, h2(s) h1(s)∞ s h1(τ)h2(τ)dτ ds < ∞
When g(r) ≡ 0, we prove that the above Cauchy problem has a nontrivial solution u(t, r) in the form u(t, r) = v(t)ω(r) for which lim t→0 u L2([0,∞)) = ∞
We study the properties of the solutions of the Cauchy problem utt − Δu gs = f (u) + g |x|, x ∈ Rn, n ≥ 2, (1)
Summary
When gs is the Minkowski metric and u0, u1 ∈ Ꮿ∞0 (R3) in [1] (see [2, Section 6.3]), it is proved that there exists T > 0 and a unique local solution u ∈ Ꮿ2([0, T) × R3) for the Cauchy problem utt − Δu gs = f (u), f ∈ Ꮿ2(R), t ∈ [0, T], x ∈ R3, u t=0 = u0, ut t=0 = u1,. When gs is the Reissner-Nordstrom metric, n = 3, p > 1, q ≥ 1, γ ∈ (0, 1) are fixed constants, f ∈ Ꮿ1(R1), f (0) = 0, a|u| ≤ f (u) ≤ b|u|, g ∈ Ꮿ(R+), g(|x|) ≥ 0, g(|x|) = 0 for |x| ≥ r1, a and b are positive constants, r1 > 0 is suitable chosen, in [6], it is proved that the initial value problem (1), (2) has nontrivial solution u ∈ Ꮿ((0, 1]Bγp,q(R+)) in the form. In the appendix we will prove some results which are used for the proof of Theorems 1.1 and 1.2
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