On the nonlinear wave equation Utt - B(t, ∥Ux∥2)Uxx = f(x, t, u, Ux, Ut) associated with the mixed homogeneous conditions
In: Journal of mathematical analysis and applications, Jg. 274 (2002), Heft 1, S. 102-123
Online
academicJournal
- print, 12 ref
In this paper we consider the following nonlinear wave equation utt - B(t, ∥ux∥2)uxx = f(x, t, u, ux, ut), x ∈ Ω = (0, 1), 0 < t < T, (1) ux (0, t) - h0u(0, t) = 0, ux (1, t) + h1u(1, t) = 0, (2) u(x, 0) = u0(x), ut(x, 0) = u1(x), (3) where h0, h1 are given nonnegative constants and B, f, u0, u1 are given functions. In Eq. (1) the coefficient B(t, ∥ux∥2) containing an integral ∥ux∥2 = f10 |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 ∈ C2(R2+), B ≥ b0 > 0, B1 ∈ C1(R2+), B1 ≥ 0, f ∈ C2(Ω x [0, ∞) x R3) and f1 ∈ C1 (Ω x [0, ∞) x R3) we obtain from the following equation utt - (B(t, ∥ux∥2) + εB1 (t, ∥ux∥2))uxx = f(x, t, u, ux, ut) + εf1 (x, t, u, ux, ut) associated to (2), (3) a weak solution uε (x, t) having an asymptotic expansion of order 2 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) associated with the mixed homogeneous conditions
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Autor/in / Beteiligte Person: | NGUYEN THANH, LONG |
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Zeitschrift: | Journal of mathematical analysis and applications, Jg. 274 (2002), Heft 1, S. 102-123 |
Veröffentlichung: | San Diego, CA: Elsevier, 2002 |
Medientyp: | academicJournal |
Umfang: | print, 12 ref |
ISSN: | 0022-247X (print) |
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