On the nonlinear wave equation utt−B(t,u2,ux2,ut2)uxx=f(x,t,u,ux,ut,u2,ux2,ut2) associated with the mixed homogeneous conditions
In: Nonlinear Analysis, Jg. 58 (2004-09-01), S. 933-959
Online
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Zugriff:
In this paper we consider the following nonlinear wave equation: (1) u tt −B(t,||u|| 2 ,||u x || 2 ,||u t || 2 )u xx =f(x,t,u,u x ,u t ,||u|| 2 ,||u x || 2 ,||u t || 2 ),x∈(0,1), 0 (2) u x (0,t)−h 0 u(0,t)=u x (1,t)+h 1 u(1,t)=0, (3) u(x,0)= u 0 (x), u t (x,0)= u 1 (x), where h0>0,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 ,||u x || 2 ,||u t || 2 ), f(x,t,u,u x ,u t ,||u|| 2 ,||u x || 2 ,||u t || 2 ) depending on the integrals ||u|| 2 = ∫ Ω |u(x,t)| 2 d x, ||u x || 2 = ∫ 0 1 |u x (x,t)| 2 d x and ||u t || 2 = ∫ Ω |u t (x,t)| 2 d x . 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∈C N+1 (R + 4 ), B⩾b 0 >0, B 1 ∈C N (R + 4 ), B 1 ⩾0, f∈C N+1 ([0,1]×R + ×R 3 ×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)+eB1(t,||u||2,||ux||2,||ut||2)]uxx=f(x,t,u,ux,ut,||u||2,||ux||2,||ut||2)+ef1(x,t,u,ux,ut,||u||2,||ux||2,||ut||2) associated to (2), (3) a weak solution ue(x,t) having an asymptotic expansion of order N+1 in e, for e sufficiently small.
Titel: |
On the nonlinear wave equation utt−B(t,u2,ux2,ut2)uxx=f(x,t,u,ux,ut,u2,ux2,ut2) associated with the mixed homogeneous conditions
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Autor/in / Beteiligte Person: | Nguyen Thanh Long |
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Zeitschrift: | Nonlinear Analysis, Jg. 58 (2004-09-01), S. 933-959 |
Veröffentlichung: | Elsevier BV, 2004 |
Medientyp: | unknown |
ISSN: | 0362-546X (print) |
DOI: | 10.1016/j.na.2004.05.021 |
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