Abstract
The limiting amplitude principle for the nonlinear Lamb system
Highlights
Consider the following problem for a real-valued function u(x, t) ∈ C(R2):(μ + mδ(x))u(x, t) = κu (x, t) + δ(x)F (u(x, t)), t ∈ R, x ∈ R. (1)Here m 0, μ, κ > 0; u ≡ ∂u/∂t, u ≡ ∂u/∂x, δ(x) is the Dirac δ–function
In the case m = 0, the string is coupled to a spring of rigidity F (y)
In the case m > 0, a ball of mass m is attached to the string at the point x = 0, and the field F (y) subjects the ball
Summary
The function up(x, t) is a solution to equation (1) for t > 0 under the condition up(x, t)|t 0 = q(x + at). Where the infinitum is taken over all solutions up(x, t) ∈ E of problem (2), (3), such that up(0±, t) = yp(t) and (yp(t), yp(t)) ∈ St. We give additional restrictions on the function F (y) (see Examples 1–3 and conditions (F1)–(F3) below), under which the set I has a unique point. Equation (18) implies the following initial condition for the function y(t): y(0) = f±(0) + g±(0) = u0(0).
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