Hyperbolic–parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates

Abstract

We consider the second order Cauchy problem εu ε + u ε + m A 1/2 u ε 2 Au ε = 0, u ε (0) = u 0 , u ε (0) = u 1 , and the first order limit problemand m : [0, +∞) → [0, +∞) is a function of class C 1 .We prove decay estimates (as t → +∞) for solutions of the first order problem, and we show that analogous estimates hold true for solutions of the second order problem provided that ε is small enough. We also show that our decay rates are optimal in many cases.The abstract results apply to parabolic and hyperbolic partial differential equations with nonlocal nonlinearities of Kirchhoff type.

The quasilinear parabolic kirchhoff equation

“…In another papers M. Ghisi and M. Gobbino [4,5] showed certain connections between the above equation and equation of hyperbolic type containing term u t t . However M. Nakao proved the existence of solutions of the equation of hyperbolic type.…”

Hyperbolic–parabolic singular perturbation for quasilinear equations of Kirchhoff type with weak dissipation

“…Hashimoto and Yamazaki [9] proved time-decay convergence estimates. Recently, Ghisi and Gobbino [10,11] showed time-global convergence estimates for mildly degenerate Kirchhoff equation.…”

“…Let c(t) be a C 1 function on [0, ∞) satisfying (9) and (10). Assume moreover that r >1 in the case p = 0.…”

“…There exist positive constants ε 0 and C depending on , and p, b 0 , b 1 , b 2 in (3) and (4), and c 0 , c 1 , c 2 , r in (9) and (10), such that the following assertions hold for every 0<ε ε 0 and , (1) be an element such that u (1) = A ũ (1) , and let…”

Optimal decay–error estimates for the hyperbolic–parabolic singular perturbation of a degenerate nonlinear equation

“…(1.6) These estimates have been obtained for the first time in [19], and then in [6] in full generality.…”