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

Abstract

We consider degenerate Kirchhoff equations with a small parameter epsilon in
front of the second-order time-derivative. It is well known that these
equations admit global solutions when epsilon is small enough, and that these
solutions decay as t -> +infinity with the same rate of solutions of the limit
problem (of parabolic type).
In this paper we prove decay-error estimates for the difference between a
solution of the hyperbolic problem and the solution of the corresponding
parabolic problem. These estimates show in the same time that the difference
tends to zero both as epsilon -> 0, and as t -> +infinity. Concerning the decay
rates, it turns out that the difference decays faster than the two terms
separately (as t -> +infinity).
Proofs involve a nonlinear step where we separate Fourier components with
respect to the lowest frequency, followed by a linear step where we exploit
weighted versions of classical energies.

Full text

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arXiv:1108.3758v1 [math.AP] 18 Aug 2011
Hyperbolic-parabolic singular perturbation for mildly
degenerate Kirchhoff equations: decay-error estimates
Marina Ghisi
Universit` a degli Studi di Pisa
Dipartimento di Matematica “Leonida Tonelli”
PISA (Italy)
e-mail: ghisi@dm.unipi.it
Massimo Gobbino
Universit` a degli Studi di Pisa
Dipartimento di Matematica Applicata “Ulisse Dini”
PISA (Italy)
e-mail: m.gobbino@dma.unipi.it

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Abstract
We consider degenerate Kirchhoff equations with a small parameter ε in front of the
second-order time-derivative. It is well known that these equations admit global solu-
tions when ε is small enough, and that these solutions decay as t → +∞ with the same
rate of solutions of the limit problem (of parabolic type).
In this paper we prove decay-error estimates for the difference between a solution of
the hyperbolic problem and the solution of the corresponding parabolic problem. These
estimates show in the same time that the difference tends to zero both as ε → 0+, and
as t → +∞. Concerning the decay rates, it turns out that the difference decays faster
than the two terms separately (as t → +∞).
Proofs involve a nonlinear step where we separate Fourier components with respect
to the lowest frequency, followed by a linear step where we exploit weighted versions of
classical energies.
Mathematics Subject Classification 2000 (MSC2000): 35B25, 35L70, 35L80.
Key words: hyperbolic-parabolic singular perturbation, quasilinear hyperbolic equa-
tions, degenerate hyperbolic equations, Kirchhoff equations, decay-error estimates.

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1 Introduction
Let H be a separable real Hilbert space. For every x and y in H, |x| denotes the norm
of x, and ?x,y? denotes the scalar product of x and y. Let A be a self-adjoint linear
operator on H with dense domain D(A). We assume that A is nonnegative, namely
?Ax,x? ≥ 0 for every x ∈ D(A), so that for every α ≥ 0 the power Aαx is defined
provided that x lies in a suitable domain D(Aα).
We consider the Cauchy problem
εu′′
ε(t) + u′
ε(t) + |A1/2uε(t)|2γAuε(t) = 0
uε(0) = u0,
∀t ≥ 0, (1.1)
u′
ε(0) = u1, (1.2)
where ε > 0 and γ ≥ 1 are real parameters, and (u0,u1) ∈ D(A) × D(A1/2) are initial
conditions satisfying the mild nondegeneracy condition
A1/2u0?= 0. (1.3)
The singular perturbation problem in its generality consists in proving the conver-
gence of solutions of (1.1), (1.2) to solutions of the first order problem
u′(t) + |A1/2u(t)|2γAu(t) = 0∀t ≥ 0,(1.4)
u(0) = u0,(1.5)
obtained setting formally ε = 0 in (1.1), and omitting the second initial condition
in (1.2). Following the approach introduced by J. L. Lions [18] in the linear case, one
defines the corrector θε(t) as the solution of the second order linear problem
εθ′′
ε(t) + θ′
ε(t) = 0∀t ≥ 0, (1.6)
θε(0) = 0,θ′
ε(0) = u1+ |A1/2u0|2γAu0=: w0.
ε(0) − u′(0), hence this corrector keeps into account
(1.7)
It is easy to see that θ′
the boundary layer due to the loss of one initial condition. Finally, one defines rε(t) and
ρε(t) in such a way that
ε(0) = u′
uε(t) = u(t) + θε(t) + rε(t) = u(t) + ρε(t)∀t ≥ 0.
With these notations, the singular perturbation problem consists in proving that
rε(t) → 0 or ρε(t) → 0 in some sense as ε → 0+.
The singular perturbation problem for Kirchhoff equations has generated a consider-
able literature in the last 30 years. The state of the art has been recently presented in the
survey [13], where more general nonlinearities and more general dissipative terms have
also been considered. In [13] the general problem has been split into six subproblems,
which we list below.
(P1) Global existence and decay estimates for the parabolic problem.
1

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(P2) Local existence for the hyperbolic problem and local-in-time error estimates on
ρε(t) and rε(t).
(P3) Global existence for the hyperbolic problem.
(P4) Decay estimates for solutions of the hyperbolic problem (as t → +∞).
(P5) Global-in-time error estimates for the singular perturbation problem, which means
time-independent estimates on ρε(t) or rε(t) as ε → 0+.
(P6) Decay-error estimates for the singular perturbation problem, which means esti-
mates such as
|Aαρε(t)| ≤ ω(ε)σ(t) or|Aαrε(t)| ≤ ω(ε)σ(t), (1.8)
where of course the convergence rate ω(ε) tends to 0 as ε → 0+, and the decay
rate σ(t) tends to 0 as t → +∞. Decay-error estimates are the meeting point of
subproblems (P4) and (P5), and they represent the ultimate goal of the theory.
Subproblem (P1) is well understood (see [1, 2, 15, 19]). The result is that problem
(1.4), (1.5) has a unique global solution for every u0∈ D(A) (and even for less regu-
lar data), and this solution decays at infinity as solutions of the ordinary differential
equation
y′+ |y|2γy = 0,
which is just the special case of (1.4) where H = R and A is the identity.
Also subproblem (P2) is well understood, because on a fixed bounded time interval
the degeneracy of the equation plays no role. Local-in-time error estimates were proved
by B. F. Esham and R. J. Weinacht in [4], then by the second author in [16], and finally
by the authors in [10, Appendix A] with optimal assumptions on initial data. The
typical result is that |A1/2ρε(t)| ≤ Cε when (u0,u1) ∈ D(A3/2)×D(A1/2), and we know
that this space is optimal if we look for estimates on |A1/2ρε(t)| of order ε, even in the
linear case (see [8]).
Subproblem (P3) was solved by K. Nishihara and Y. Yamada [22]. They proved that
(1.1), (1.2) has a unique global solution provided that (u0,u1) ∈ D(A)×D(A1/2) satisfy
the nondegeneracy assumption (1.3) and ε is small enough. It is not known whether
the smallness of ε is a necessary condition. This remains the main open problem in
the theory of Kirchhoff equations, both dissipative and non-dissipative, both degenerate
and non-degenerate.
Subproblem (P4) was first addressed in [22]. More recently, the authors in [9] and [6]
provided optimal decay estimates, showing that solutions of (1.1), (1.2) decay with the
same rate of solutions of the corresponding parabolic problem (see also [20, 21, 23, 24]
for the case γ = 1). The results have been recently extended in [12] to equations with
weak dissipation, namely with a dissipative term of the form b(t)u′
as t → +∞.
(1.9)
ε(t), where b(t) → 0
2

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Subproblem (P5) was considered by the authors in [10], with non-optimal conver-
gence rates, and finally by the first author [5] with optimal convergence rates.
For the convenience of the reader, in section 2.1 we state all previous results needed
in the sequel.
In this paper we concentrate on subproblem (P6), namely on decay-error estimates.
Estimates of this type were proved by R. Chill and A. Haraux [3] in the case of linear
equations, and then by H. Hashimoto and T. Yamazaki [17] for nondegenerate Kirchhoff
equations. Those results were successively extended by T. Yamazaki [26, 27] and by
the authors [11] to nondegenerate Kirchhoff equations with weak dissipation. The non-
degenerate character of the equation (namely strict hyperbolicity) seems to be essential
in all previous approaches, which fail when applied to degenerate equations. This is the
technical reason why subproblem (P6) resisted so far as an open problem.
In this paper we begin by showing that there is a deeper reason. Indeed we show in
Example 2.2 that, without further assumptions on initial data, the expected decay-error
estimates are actually false, even in the simple case where H is a two dimensional vector
space. By “expected” we mean decay-error estimates such as (1.8), where the decay-rate
σ(t) is the same as in subproblem (P4), and the convergence rate ω(ε) is the same as in
subproblem (P2) or subproblem (P5). The rigorous verification of the counterexamples
strongly relies on the asymptotic limits which have been recently found in [6].
Roughly speaking, the expected decay-error estimates are false whenever the initial
condition u1has a nonzero Fourier component with respect to a frequency which is less
than all frequencies corresponding to nonzero components of u0. This motivates the in-
troduction of a special class of initial data where this cannot happen (see Definition 2.3).
In Remark 2.4 we show that this requirement on initial data is easily satisfied in many
concrete cases.
The main result of this paper is that in this class of initial data we do have decay-error
estimates for the degenerate problem. Apart from the special assumption, the regularity
we require on initial data is optimal, because it is the same which was optimal in the
linear nondegenerate case. The convergence rates ω(ε) are optimal, because they are
the same which appear in the local-in-time error estimates of subproblem (P2), or in the
global-in-time error estimates of subproblem (P5). The real surprise lies in the decay
rate σ(t). Indeed it turns out that ρε(t) and rε(t) decay faster than uε(t) and u(t) alone.
An improvement of decay rates has been observed also in [3] and [17, 26], but in
those cases it seems to originate from different reasons. Indeed in those examples it
is essential that the operator is not coercive, while in our case we have improvement
even if the operator is coercive. Roughly speaking, our improvement comes from the
fact that our equation is in the same time degenerate and nonlinear. In section 2.3
below we show a simple toy model, based on ordinary differential equations of order
one, which gives a flavor of this aspect. The main point, both for the improvement and
for the impossibility of expected decay-error estimates for general data, is that solutions
of (1.9) decay as C(1+t)−1/(2γ), where the constant C depends on γ, but is independent
of the initial condition.
Our result requires a new approach in order to take advantage of the special assump-
3