Linear Recursive Schemes and Asymptotic Expansion Associated with the Kirchoff–Carrier Operator

Abstract

In this paper we consider a nonlinear wave equation with the Kirchoff-Carrier operator,where b 0 > 0 is a given constant and B f are the given functions. In Eq. (1) the function B ∇u 2 depends on the integral ∇u 2 = ∇u 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 the standard compactness argument. If B ∈ C 2 R + B 1 ∈ C 1 R + f ∈ C 2 × 0 ∞ × R 3 , and f 1 ∈ C 1 × 0 ∞ × R 3 then an asymptotic expansion of order 2 in ε is obtained with a right-hand side of the form f x t u u x u t + εf 1 x t u u x u t , and B stands for B + εB 1 , for ε sufficiently small.  2002 Elsevier Science (USA)

Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions

“…4. This result is a relative generalization of [1,5,6,8,9,[12][13][14][15][16][17][18][23][24][25][26][27][28][29][30][31].…”

“…(ii) In the case of μ = μ( u x 2 ) or μ = μ(u), f = f (x, t, u, u x , u t ) with f ∈ C 1 ([0, 1]× R + × R 3 ), and some other boundary conditions standing for (1.2), we have also obtained some results in the papers [13,16,24].…”

Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition

“…It is proved that under these conditions, a unique solution u(t) exists on R + such that u / (t) + u x (t) decay exponentially to 0 as t → +∞. The results obtained here relatively are in part generalizations of those in [1][2][3][6][7][8][9][10]. Finally, we present some numerical results.…”

On the nonlinear wave equation Utt−B(t,‖Ux‖2)Uxx=f(x,t,U,Ux,Ut,‖Ux‖2) associated with the mixed nonhomogeneous conditions

“…If B ∈ C 2 (R + ), B 1 ∈ C 1 (R + ), B 0, B 1 0, f ∈ C 2 (Ω × R + × R 3 ) and f 1 ∈ C 1 (Ω × R + × R 3 ), we have obtained an asymptotic expansion of the weak solution u ε of order 2 in ε, for ε sufficiently small [7]. In this paper we shall first associate with the problem (1.1)-(1.3) a linear recurrent sequence which is bounded in a suitable space of functions.…”

“…In [5] the authors have studied the existence and uniqueness of the equation u tt + λ∆ 2 u − B ∇u 2 ∆u + ε|u t | α−1 u t = F (x, t), x ∈ Ω, t > 0, (1.6) where λ > 0, ε > 0, 0 < α < 1, are given constants, and Ω is a bounded open set of R n . In [7] we have studied the linear recursive schemes and asymptotic expansion associated with the nonlinear wave equation (x, t, u, u x , u t ) + εf 1 (x, t, u, u x , u t ),…”