UNIFORM CONTINUITY OF THE SOLUTION MAP FOR NONLINEAR WAVE EQUATION IN REISSNER-NORDSTRÖM METRIC

In this paper we study the properties of the solutions to the Cauchy problem (1) (utt −∆u)gs = f(u) + g(|x|), t ∈ [0, 1], x ∈ R, (2) u(1, x) = u0 ∈ Ḣ(R), ut(1, x) = u1 ∈ L(R), where gs is the Reissner-Nordstrom metric (see [2]); f ∈ C1(R1), f(0) = 0, a|u| ≤ f (u) ≤ b|u|, g ∈ C(R+), g(|x|) ≥ 0, g(|x|) = 0 for |x| ≥ r1, a and b are positive constants, r1 > 0 is suitable chosen. When g(r) ≡ 0 we prove that the Cauchy problem (1), (2) has a nontrivial solution u(t, r) in the form u(t, r) = v(t)ω(r) ∈ C((0, 1]Ḣ1(R+)), where r = |x|, and the solution map is not uniformly continuous. When g(r) 6= 0 we prove that the Cauchy problem (1), (2) has a nontrivial solution u(t, r) in the form u(t, r) = v(t)ω(r) ∈ C((0, 1]Ḣ1(R+)), where r = |x|, and the solution map is not uniformly continuous. Subject classification: Primary 35L10, Secondary 35L50.

When g s is the Minkowski metric and initial data are in C ∞ 0 (R 3 ), in [4](see and [1], section 6.3) is proved that there exists a number 0 > 0 such that for any data (u 0 , u 1 ) ∈ C ∞ 0 (R 3 ) with E(u(0)) < 0 , the initial value problem admits a global smooth solution.When g s is the Minkowski metric in [6] is proved that the Cauchy problem a and b are positive constants, r 1 > 0 is suitable chosen, in [7] is proved that the initial value problem (1), (2), has nontrivial solution u ∈ C((0, 1] Ḃγ p,q (R + )) in the form where r = |x|, for which lim t−→0 ||u|| Ḃγ p,q (R + ) = ∞.In this paper we will prove that the Cauchy problem (1), (2) has nontrivial solution u = u(t, r) ∈ C((0, 1] Ḣ1 (R + )) and the solution map is not uniformly continuous.When we say that the solution map (u • , u 1 , g) −→ u(t, r) is uniformly continuous we understand: for every positive constant there exist positive constants δ and R such that for any two solutions u, v of the Cauchy problem (1), (2), with right hands g = g 1 , g = g 2 of (1), so that the following inequality holds where Our main results are Theorem 1.1.Let K, Q are positive constants for which The paper is organized as follows.In section 2 we prove theorem 1.1.In section 3 we prove theorem 1.2.
There exists function v(t) for which (H1)-(H3) are hold.Really, let us consider the function , where the constants A and a satisfy the conditions = 0, consequently (H2) is hold.On the other hand we have i.e. (H3) is hold.
EJQTDE, 2007 No. 12, p. 6 In this section we will prove that the homogeneous Cauchy problem (1), ( 2) has nontrivial solution in the form Let us consider the integral equation ( ) where u(t, r) = v(t)ω(r).

Uniformly continuity of the solution map for the homogeneous Cauchy problem (1), (2)
Let v(t) is same function as in Theorem 2.1.

3. 2 .
Uniformly continuity of the solution map for the nonhomogeneous Cauchy problem (1),(2) Let v(t) is same function as in Theorem 3.1.