Current Research in Nonlinear Analysis by Themistocles M. Rassias

Author:Themistocles M. Rassias
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

where F n (t, x) = μ n (t, (−∞, x]) for all and F(t, x) = μ(t, (−∞, x]).

Moreover, for each , (u n (t, ⋅), ρ n (t, ⋅), μ n (t, ⋅)) is a smooth solution to the 2CH system, that is, u n (t, x) and ρ n (t, x) belong to and for all t ≥ 0, and, in particular, no wave breaking occurs.

Proof

Since we showed in Theorem 3 that X n  = (y n , U n , h n , r n ) converges to X = (y, U, h, 0) in E, and hence according to Theorem 4, the sequence (u n,0, ρ n,0, μ n,0) converges to (u 0, 0, μ 0) in the sense of Definition 5, the first part of the theorem is an immediate consequence of Theorem 6.

As far as the smoothness of the solution (u n (t, ⋅), ρ n (t, ⋅), μ n (t, ⋅)) for any t ≥ 0 and is concerned, we refer the interested reader to [16, Sect. 6].

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