On the nonlinear wave equation utt−B(t,‖u‖2,‖ux‖2)uxx=f(x,t,u,ux,ut,‖u‖2,‖ux‖2) associated with the mixed homogeneous conditions
In: Journal of Mathematical Analysis and Applications, , Heft 1, S. 243-268
Online
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Zugriff:
In this paper we consider the following nonlinear wave equation: (1)utt−B(t,‖u‖2,‖ux‖2)uxx=f(x,t,u,ux,ut,‖u‖2,‖ux‖2), x∈(0,1), 00, h1⩾0 are given constants and B, f, u˜0, u˜1 are given functions. In Eq. (1), the nonlinear terms B(t,‖u‖2,‖ux‖2), f(x,t,u,ux,ut,‖u‖2,‖ux‖2) depend on the integrals ‖u‖2=∫Ω|u(x,t)|2dx and ‖ux‖2=∫01|ux(x,t)|2dx. In this paper I 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∈CN+1(R+3), B⩾b0>0, B1∈CN(R+3), B1⩾0, f∈CN+1([0,1]×R+×R3×R+2) and f1∈CN([0,1]×R+×R3×R+2) we obtain for the following equation utt−[B(t,‖u‖2,‖ux‖2)+ɛB1(t,‖u‖2,‖ux‖2)]uxx=f(x,t,u,ux,ut,‖u‖2,‖ux‖2)+ɛf1(x,t,u,ux,ut,‖u‖2,‖ux‖2) associated to (2), (3) a weak solution uɛ(x,t) having an asymptotic expansion of order N+1 in ɛ, for ɛ sufficiently small.
Titel: |
On the nonlinear wave equation utt−B(t,‖u‖2,‖ux‖2)uxx=f(x,t,u,ux,ut,‖u‖2,‖ux‖2) associated with the mixed homogeneous conditions
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Autor/in / Beteiligte Person: | Long, Nguyen Thanh |
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Zeitschrift: | Journal of Mathematical Analysis and Applications, , Heft 1, S. 243-268 |
Veröffentlichung: | Elsevier Inc. |
Medientyp: | unknown |
ISSN: | 0022-247X (print) |
DOI: | 10.1016/j.jmaa.2004.12.053 |
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