Abstract
This paper is devoted to the study of a nonlinear Carrier wave equation in an annular membrane associated with Robin-Dirichlet conditions. Existence and uniqueness of a weak solution are proved by using the linearization method for nonlinear terms combined with the Faedo-Galerkin method and the weak compact method. Furthermore, an asymptotic expansion of a weak solution of high order in a small parameter is established.
Highlights
In this paper, we consider the following nonlinear Carrier wave equation in the annular membrane: utt −μ (‖u (t)‖20) =f (x, t, u, ux, ut) (1)ρ < x < 1, 0 < t < T, associated with Robin-Dirichlet conditions u (ρ, t) = ux (1, t) + ζu (1, t) = 0, (2)and initial conditions u (x, 0) = ũ0 (x), (3)
Our results can be regarded as an extension and improvement of the corresponding results of [15, 16]
Notation ‖ ⋅ ‖ stands for the norm in L2 and we denote ‖ space of the We norm in Banach space X
Summary
We consider the following nonlinear Carrier wave equation in the annular membrane: utt. Equation (1) is the bidimensional nonlinear wave equation describing nonlinear vibrations of annular membrane Ω1 = {(x, y) : ρ2 < x2 + y2 < 1}. The area of the annular membrane and the tension at various points change in time. The condition on boundary Γ1 = {(x, y) : x2 + y2 = 1}, that is, ux(1, t) + ζu(1, t) = 0, describes elastic constraints where ζ constant has a mechanical signification. With the boundary condition on Γρ = {(x, y) : x2 +y2 = ρ2} requiring u(ρ, t) = 0, the annular membrane is fixed. In [1], Carrier established the equation which models vibrations of an elastic string when changes in tension are not small: ρutt Our results can be regarded as an extension and improvement of the corresponding results of [15, 16]
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