On the nonlinear wave equation <f>u<subscript>tt</subscript>-B(t,||u||<superscript>2</superscript>,||u<subscript>x</subscript>||<superscript>2</superscript>,||u<subscript>t</subscript>||<superscript>2</superscript>)u<subscript>xx</subscript>=f(x,t,u,u<subscript>x</subscript>,u<subscript>t</subscript>,||u||<superscript>2</superscript>,||u<subscript>x</subscript>||<superscript>2</superscript>,||u<subscript>t</subscript>||<superscript>2</superscript>)</f> associated with the mixed homogeneous conditions
In: Nonlinear Analysis, Jg. 58 (2004-09-01), Heft 7/8, S. 933-959
Online
academicJournal
In this paper we consider the following nonlinear wave equation: where h0>0,h1⩾0 are given constants and B,f, u 0, u 1 are given functions. In , the nonlinear terms B(t,||u||2,||ux||2,||ut||2), f(x,t,u,ux,ut,||u||2,||ux||2,||ut||2) depending on the integrals ||u||2=∫lower limit Ω |u(x,t)|2 dx, ||ux||2=∫lower limit 0, upper limit 1 |ux(x,t)|2 dx and ||ut||2=∫lower limit Ω |ut(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∈CN+1(R+4), B⩾b0>0, B1∈CN(R+4), B1⩾0, f∈CN+1([0,1]×R+×R3×R+3) and f1∈CN([0,1]×R+×R3×R+3) we obtain from the following equation utt-[B(t,||u||2,||ux||2,||ut||2)+#x03B5;B1(t,||u||2,||ux||2,||ut||2)]uxx=f(x,t,u,ux,ut,||u||2,||ux||2,||ut||2)+#x03B5;f1(x,t,u,ux,ut,||u||2,||ux||2,||ut||2) associated to (2), (3) a weak solution u#x03B5;(x,t) having an asymptotic expansion of order N+1 in #x03B5; , for #x03B5; sufficiently small. [Copyright &y& Elsevier]
Titel: |
On the nonlinear wave equation <f>u<subscript>tt</subscript>-B(t,||u||<superscript>2</superscript>,||u<subscript>x</subscript>||<superscript>2</superscript>,||u<subscript>t</subscript>||<superscript>2</superscript>)u<subscript>xx</subscript>=f(x,t,u,u<subscript>x</subscript>,u<subscript>t</subscript>,||u||<superscript>2</superscript>,||u<subscript>x</subscript>||<superscript>2</superscript>,||u<subscript>t</subscript>||<superscript>2</superscript>)</f> associated with the mixed homogeneous conditions
|
---|---|
Autor/in / Beteiligte Person: | Long, Nguyen Thanh |
Link: | |
Zeitschrift: | Nonlinear Analysis, Jg. 58 (2004-09-01), Heft 7/8, S. 933-959 |
Veröffentlichung: | 2004 |
Medientyp: | academicJournal |
ISSN: | 0362-546X (print) |
DOI: | 10.1016/j.na.2004.05.021 |
Schlagwort: |
|
Sonstiges: |
|