On the nonlinear wave equation Utt - B(t, ∥Ux∥2)Uxx = f(x, t, u, Ux, ut, ∥Ux∥2) associated with the mixed nonhomogeneous conditions
In: Journal of mathematical analysis and applications, Jg. 292 (2004), Heft 2, S. 433-458
Online
academicJournal
- print, 13 ref
In this paper we consider the following nonlinear wave equation utt - B(t, ∥ux∥2)uxx = f(x,t,u,ux,ut,∥ux∥2), x ∈ Ω = (0, 1), 0 < t < T, (1) ux(0,t) -h0u(0,t) = g0 (t), u(1, t) = g1(t), (2) u(x, 0) = μ0(x), ut(x, 0) = μ1(x), (3) where B, f, g0, g1, μ0, μ1 are given functions. In Eq. (1), the nonlinear terms B(t, ∥ux∥2), f(x, t, u, ux, ut, ∥ux∥2) depending on an integral ∥ux∥2 = ∫01 |ux(x, t)|2 dx. In this paper we associate with problem (1)-(3) a linear recursive scheme for which the existence of a local and unique solution is proved by using standard compactness argument. In case of B ∈ C3(R2+), B ≥ b0 > 0, B1 ∈ C2(R2+), B1 ≥ 0, f ∈ C3([0, 1] x R+ × R3 x R+) and f1 ∈ C2([0, 1] x R+ x R3 x R+) we obtain from the equation utt - [B(t, ∥ux∥2)+ΣB1(t,∥ux∥2)]uxx = f(x,t,u,ux,ut,∥ux∥2) + εft(x,t,u, ux,ut, ∥ux∥2) associated to (2), (3) a weak solution uε(x, t) having an asymptotic expansion of order 3 in ε, for ε sufficiently small.
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On the nonlinear wave equation Utt - B(t, ∥Ux∥2)Uxx = f(x, t, u, Ux, ut, ∥Ux∥2) associated with the mixed nonhomogeneous conditions
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Autor/in / Beteiligte Person: | NGUYEN THANH, LONG ; BUI TIEN, DUNG |
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Zeitschrift: | Journal of mathematical analysis and applications, Jg. 292 (2004), Heft 2, S. 433-458 |
Veröffentlichung: | San Diego, CA: Elsevier, 2004 |
Medientyp: | academicJournal |
Umfang: | print, 13 ref |
ISSN: | 0022-247X (print) |
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