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, Jg. 306 (2005-06-01), S. 243-268
Online
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Zugriff:
In this paper we consider the following nonlinear wave equation (1) u tt −B t,‖u x ‖ 2 u xx =f x,t,u,u x ,u t ,‖u x ‖ 2 , x∈Ω=(0,1), 0 (2) u x (0,t)−h 0 u(0,t)=g 0 (t), u(1,t)=g 1 (t), (3) u(x,0)= u 0 (x), u t (x,0)= u 1 (x), where B,f,g 0 ,g 1 , u 0 , u 1 are given functions. In Eq. (1), the nonlinear terms B ( t ,‖ u x ‖ 2 ), f ( x , t , u , u x , u t ,‖ u x ‖ 2 ) depending on an integral ‖u x ‖ 2 =∫ 0 1 |u x (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 ∈ C 3 ( R + 2 ), B ⩾ b 0 >0, B 1 ∈ C 2 ( R + 2 ), B 1 ⩾0, f ∈ C 3 ([0,1]× R + × R 3 × R + ) and f 1 ∈ C 2 ([0,1]× R + × R 3 × R + ) we obtain from the equation u tt −[ B ( t ,‖ u x ‖ 2 )+ eB 1 ( t ,‖ u x ‖ 2 )] u xx = f ( x , t , u , u x , u t ,‖ u x ‖ 2 )+ ef 1 ( x , t , u , u x , u t ,‖ u x ‖ 2 ) associated to (2), (3) a weak solution u e ( x , t ) having an asymptotic expansion of order 3 in e , for e 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: | Nguyen Thanh Long |
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Zeitschrift: | Journal of Mathematical Analysis and Applications, Jg. 306 (2005-06-01), S. 243-268 |
Veröffentlichung: | Elsevier BV, 2005 |
Medientyp: | unknown |
ISSN: | 0022-247X (print) |
DOI: | 10.1016/j.jmaa.2004.12.053 |
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