Scattering for quasilinear hyperbolic equations of Kirchhoff type with perturbation

Abstract

This paper is concerned with the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation. We show the existence of the wave operators and the scattering operator for small data, and that these operators are homeomorphic with respect to a suitable metric in a neighborhood of the origin. Introduction Let H be a separable complex Hilbert space H with the inner product (·, ·)H and the norm ‖ · ‖. Let A be a non-negative injective self-adjoint operator with domain D(A), and let m be a function satisfying m ∈ C2([0,∞); [m0,∞)), with a positive constant m0. Let b(t) be a C1 function on R. We consider the initial value problem of the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation u′′(t) + b(t)u′(t) + m(‖A1/2u(t)‖2)2Au(t) = 0, (0.1) u(0) = φ0, u′(0) = ψ0. (0.2) The asymptotic behavior of the solution of the equation above depends on the integrability of b with respect to t. If b(t) = (1 + t)−p with 0 ≤ p ≤ 1, it is known that the global solution of (0.1)-(0.2) exists uniquely and behaves like solutions of a corresponding parabolic equation, for small initial data (φ0, ψ0) ∈ D(A) × D(A1/2) (see Yamazaki [14] for 0 ≤ p < 1 and Ghisi and Gobbino [7] for p = 1, and see Ghisi [6] for a mildly degenerate case m(λ) = λ with γ ≥ 1 and 0 ≤ p ≤ 1). There are no result about the global solvability for large initial data in Sobolev spaces, even for the constant dissipation term. On the other hand, if b satisfies the assumptions lim t→±∞ b(t) = 0, (0.3) 〈t〉ptb′(t) ∈ L1(R), (0.4) where p ≥ 0 is a constant and 〈t〉 := (1+ |t|2)1/2, the author [15] showed the global existence of a solution for small data in some class and showed the following (see Theorems B and C): 2010 Mathematics Subject Classification. Primary 35L72; Secondary 35L90.