The Geometric Optics for the Kirchhoff-type equations

Karen Yagdjian

Institute of Mathematics, National Acad. Sci. of Armenia, Yerevan, 375019, Armenia

Abstract

We study the nonlinear first-order symmetric hyperbolic integro-differential system on ℝ_{t} × ℝ_{x}^{d}

\frac{∂U}{∂t} − ∑_{j=1}^{d} A_{j}(s(t))\frac{∂U}{∂x_{j}} + s′(t)B(s(t))U = G(t, x) ,  (0.1)

where A_{j}(s), B(s), are N × N matrices with suitably smooth elements for s ∈ ℝ in a neighbourhood of 0. The function s(t) is given by

s(t) = ∫_{ℝ_{x}^{d}} < SU(t, x), \overline{U(t, x)} > dx , s′(t) := ds(t)/dt ,

where S = {s_{ij}} is a constant self-adjoint matrix.

The main purpose is to present a method for finding small-amplitude, high-frequency wave solutions of the Kirchhoff-type hyperbolic systems (0.1) of non-local quasilinear partial differential equations.

¿From the geometric optics point of view the basic feature of the Kirchhoff equation is a globally existing phase function. Indeed, if the initial data for the phase function is a linear function, then the eikonal equation has solution defined for all t ≥ 0. This allows to consider semi-global weakly nonlinear geometric optics approximation, that is an approximation which is applicable for every given time-interval [0, T], T > 0. The transport equation is ℝ-linear and nonlocal. It has solution existing in general on the bounded interval [0, T].

We construct the formal solutions of (0.1) with expansions of the form

u(ε, t, x) = u_{0}(t, x) + U(ε, t, x, ϕ(t, x)/ε) ,

where u_{0}(t, x) is a given solution to the equation, ϕ is a phase function, while U(ε, t, x, θ) is periodic in θ and is constructed as an asymptotic series

U(ε, t, x, θ) ∼ ∑_{j=1}^{∞} ε^{j}U_{j}(t, x, θ) .

To justify nonlinear geometric optics for the system (0.1) we establish results on a global solvability and on a global stability with respect to the right-hand side G and the initial data given at t = 0.