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

Abstract

SUMMARYWe consider a hyperbolic-parabolic singular perturbation problem for a quasilinear hyperbolic equation of Kirchhoff type with dissipation weak in time. The purpose of this paper is to give time-decay convergence estimates of the difference between the solutions of the hyperbolic equation above and those of the corresponding parabolic equation, together with the unique existence of the global solutions of the hyperbolic equation above.

The hyperbolic-elliptic equation with the nonlocal condition

“…Using the last formula and the estimates (3), (4), (5) and (6), we can obtain Now, we will obtain the same estimates for the norm of u.t/, A 1=2 u.t/ and Au.t/. Using the identities (13) and (14) and the estimates (3), (4), (5), (6), we obtain…”

“…They also play a very important role for mathematical modelling in many branches of science, engineering and industry. Theory and numerical methods of solutions of the boundary value problems endowed with the local and nonlocal boundary conditions for partial differential equations have been investigated by many researchers ( [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein).…”

On the Parabolic Regime of a Hyperbolic Equation with Weak Dissipation: The Coercive Case

“…These results have been extended to Kirchhoff equation by H. Hashimoto and T. Yamazaki [10], T. Yamazaki [15,16] and the authors [5], in the following sense.…”

“…This is the parabolic regime, with the so-called effective dissipation.• When p = 1, the dissipation is still effective (namely the integral of the coefficient diverges), but according to [13] "the parabolic asymptotics changes to a wave type asymptotics". In any case, solutions keep on going to 0, at least when ε is small enough, and for this reason the case p = 1 eventually falls in the parabolic regime.These results have been extended to Kirchhoff equation by H. Hashimoto and T. Yamazaki [10], T. Yamazaki [15,16] and the authors [5], in the following sense.• When p ∈ [0, 1], problem (1.1), (1.2) has a unique global solution provided that ε is small enough, and this solution decays to 0 as t → +∞. This is the parabolic regime.…”

Necessary condition on the dissipative term for the asymptotically free property of dissipative Kirchhoff equations

“…If b(t)u ′ (t) is effectively dissipative such as b(t) = (1 + t) −q with 0 ≤ q ≤ 1, then the global solution of (0.2)-(0.3) exists uniquely and approaches to solutions of a corresponding parabolic equation as t → ∞ for small initial data (φ 0 , ψ 0 ) ∈ D(A) × D(A 1/2 ) (see [17] for 0 ≤ q < 1, and Ghisi and Gobbino [6] for q = 1). On the other hand, if b(t) is integrable, Ghisi and Gobbino brought up the problem that non-trivial solutions of (0.2) have hyperbolic property such as they do not decay to 0 as t → ∞ provided the global solution exists (which is an open problem) (see [6]).…”