Abstract
Abstract Abstract. The aim of this paper is to analyze the properties of the solution map to the Cauchy problem for the wave map equation with a source term, when the target is the hyper- boloid H 2 that is embedded in R 3 . The initial data are in _ H 1 L 2 . We prove that the solution map is not uniformly continuous. Abstract Subject classication: Primary 35L10, Secondary 35L50. In this paper we study the properties of the solution map (u ;u1;g) ! u(t;x) to the Cauchy problem
Highlights
On the other hand (here we use (11) and the fact that sinhx increases for every x)
In this paper we study the properties of the solution map (u◦, u1, g) −→ u(t, x) to the Cauchy problem utt − u − (|ut|2 − |∇xu|2)u = g(t, x), (2)
When we say that the solution map (u◦, u1, g) −→ u(t, x) is uniformly continuous we understand: for every positive constant there exist positive constants δ and R such that for any two solutions u, v : R × R2 −→ H2 of (1), (2), with right hands g = g1, g = g2 of (1), so that (3) E(0, u − v) ≤ δ, ||g1||L1([0,1]L2(R2)) ≤ R, ||g2||L1([0,1]L2(R2)) ≤ R, ||g1 − g2||L1([0,1]L2(R2)) ≤ R, EJQTDE, 2003 No 18, p.1 the following inequality holds
Summary
On the other hand (here we use (11) and the fact that sinhx increases for every x). (here we use (11) and the fact that the functions sinhx, coshx are increasing for every x ≥ 0). In this paper we study the properties of the solution map (u◦, u1, g) −→ u(t, x) to the Cauchy problem utt − u − (|ut|2 − |∇xu|2)u = g(t, x), (2) We prove that the solution map (u◦, u1, g) −→ u(t, x) to the Cauchy problem (1), (2) is not uniformly continuous.
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