On a shock problem involving a nonlinear viscoelastic bar
- PDF / 714,836 Bytes
- 22 Pages / 468 x 680 pts Page_size
- 66 Downloads / 210 Views
We treat an initial boundary value problem for a nonlinear wave equation utt − uxx + K |u|α u + λ|ut |β ut = f (x,t) in the domain 0 < x < 1, 0 < t < T. The boundary condition at the boundary point x = 0 of the domain for a solution u involves a time convolution term of the boundary value of u at x = 0, whereas the boundary condition at the other boundary point is of the form ux (1,t) + K1 u(1,t) + λ1 ut (1,t) = 0 with K1 and λ1 given nonnegative constants. We prove existence of a unique solution of such a problem in classical Sobolev spaces. The proof is based on a Galerkin-type approximation, various energy estimates, and compactness arguments. In the case of α = β = 0, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution (u,P) of this problem up to order N + 1 in two small parameters K, λ. 1. Introduction Given T > 0, we consider the problem to find a pair of functions (u,P) such that
utt − uxx + F u,ut = f (x,t),
0 < x < 1, 0 < t < T,
ux (0,t) = P(t), ux (1,t) + K1 u(1,t) + λ1 ut (1,t) = 0, u(x,0) = u0 (x),
(1.1)
ut (x,0) = u1 (x),
where • F(u,ut ) = K |u|α u + λ|ut |β ut , • u0 , u1 , f are given functions, • K, K1 , α, β, λ and λ1 ≥ 0 are given constants
and the unknown function u(x,t) and the unknown boundary value P(t) satisfy the following Cauchy problem for ordinary differential equation P // (t) + ω2 P(t) = hutt (0,t), P(0) = P0 , Copyright © 2006 Hindawi Publishing Corporation Boundary Value Problems 2005:3 (2005) 337–358 DOI: 10.1155/BVP.2005.337
/
0 < t < T,
P (0) = P1 ,
(1.2)
338
On a shock problem involving a nonlinear viscoelastic bar
where ω > 0, h ≥ 0, P0 , P1 are given constants. Problem (1.1)–(1.2) describes the shock between a solid body and a nonlinear viscoelastic bar resting on a viscoelastic base with nonlinear elastic constraints at the side, constraints associated with a viscous frictional resistance. In [1], An and Trieu studied a special case of problem (1.1)–(1.2) with α = β = 0 and f , u0 , u1 and P0 vanishing, associated with the homogeneous boundary condition u(1,t) = 0 instead of (1.1)3 being a mathematical model describing the shock of a rigid body and a linear visoelastic bar resting on a rigid base. From (1.2), solving the equation ordinary differential of second order, we get P(t) = g(t) + hu(0,t) −
t 0
k(t − s)u(0,s)ds,
(1.3)
where
1 P1 − hu1 (0) sinωt, ω k(t) = hω sinωt.
g(t) = P0 − hu0 (0) cosωt +
(1.4)
This observation motivates to consider problem (1.1) with a more general boundary term of the form P(t) = g(t) + hu(0,t) −
t 0
k(t − s)u(0,s)ds,
(1.5)
which we will do henceforth. In [9, 10], Dinh and Long studied problem (1.1)1,2,4 and (1.5) with Dirichlet boundary condition at boundary point x = 1 in [10] extending an earlier result of theirs for k = 0 in [9]. In [15], Santos has studied the following problem utt − µ(t)uxx = 0, t
u(1,t) +
0
0 < x < 1, t > 0,
u(0,t) = 0, G(t − s)µ(s)ux (1,s)ds = 0,
u(x,0) = u0 (x),
(1.6)
ut (x,0) = u1 (x).
The integral in (1.6)3 is a boundary
Data Loading...
