STABILITY OF SOLUTIONS FOR NONLINEAR WAVE EQUATIONS WITH A POSITIVE–NEGATIVE DAMPING

AIMS’ Journals

Volume X, Number 0X, XX 200X

STABILITY OF SOLUTIONS FOR NONLINEAR WAVE EQUATIONS WITH A POSITIVE–NEGATIVE DAMPING

GENNI FRAGNELLI AND DIMITRI MUGNAI

Genni Fragnelli

Dipartimento di Ingegneria dell’Informazione Universit` a di Siena, Via Roma 56, 53100 Siena - Italy

Dimitri Mugnai

Dipartimento di Matematica e Informatica Universit` a di Perugia, Via Vanvitelli 1, 06123 Perugia - Italy

(Communicated by the associate editor name)

Abstract. We prove a stability result for damped nonlinear wave equations, when the damping changes sign and the nonlinear term satisfies a few natural assumptions.

1. Introduction and motivations. We are concerned with some classes of non- linear abstract damped wave equations, whose prototype is the usual wave equation in a bounded domain Ω ⊂ R ^N , N ≥ 1,







u tt − ∆u + h(t)u t = f (u) in (0, +∞) × Ω,

u(t, x) = 0 in (0, +∞) × ∂Ω,

u(0, x) = u 0 (x), u t (0, x) = u 1 (x) x ∈ Ω,

(1)

though we can handle equations in a more general Banach setting like u ⁰⁰ + B(t)u ⁰ + Au = f (u),

with B and A suitably given (see Section 2 for the precise setting).

The inspiring problem is, for p ≥ 0,







u _tt − ∆u + h(t)u _t = −|u| ^p u in (0, +∞) × Ω,

u(t, x) = 0 in (0, +∞) × ∂Ω,

u(0, x) = u 0 (x), u t (0, x) = u 1 (x) x ∈ Ω,

(2)

which is the counterpart of the one studied by Levine, Park and Serrin in [9],







u _tt − ∆u + h(t)u t = |u| ^p u in (0, +∞) × Ω,

u(t, x) = 0 in (0, +∞) × ∂Ω,

u(0, x) = u ₀ (x), u _t (0, x) = u ₁ (x) x ∈ Ω,

(3)

2000 Mathematics Subject Classification. Primary:35L70; Secondary: 93D20, 25B35.

Key words and phrases. damped nonlinear wave equations, positive–negative damping.

The research of G.F. is supported by the M.I.U.R. project Metodi di viscosit` a, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari. The research of D.M. is supported by the M.I.U.R. project Metodi Variazionali ed Equazioni Differenziali Nonlineari.

1

where, of course, the only difference is that they have f (u) = |u| ^p u, instead of our f (u) = −|u| ^p u. We recall that in [9] the authors prove that solutions of (3) may blow up in finite time if h ≥ 0.

In this paper we have in mind (2) with a kind of intermittent damping h which changes sign, so that it is positive in a sequence of intervals diverging to infinity and negative in another sequence of intervals diverging to infinity. In these latter regions the energy increases, so that the standard energy method is not useful at all to derive stability results. However, we show that this approach is still useful if the damping satisfies some structure condition, for example if the set of interval where it is negative is not too large if compared to the set where it is positive (see the precise general condition (18) below). The interest in intermittent time–

dependent damping goes back to the nineties, for example with the papers [15]

and [6], where the authors were interested in ordinary differential equations (or systems). Then, Haraux, Martinez and Vancostenbole in [5] considered this kind of time–dependent damping for the linear wave equation and provided some sufficient conditions that ensure asymptotic solutions of every solutions. In this paper we consider nonlinear wave equations controlled by sign–changing distributed damping and we give sufficient conditions for asymptotic stability to holds. However, as in the linear case (see [5]) and in the nonlinear case but with nonnegative damping (see [3]), we are not able to show that the required condition for stability is also necessary.

Moreover, as Pucci and Serrin in [15], we do not prove any existence result under our general assumptions on the nonlinearity, but we assume that a global solution exists. On the other hand, some existence results can be shown in special cases:

for example, for the linear homogeneous problem studied in [5], an existence result is exhibited by using standard arguments on Lipschitz perturbations of contraction semigroups.

Analogue stability results were obtained, for example, in [10], where stability is proved for the following linear problem:







u tt − ∆u + h(x)u t + k(x)u = 0 in (0, +∞) × Ω,

u(t, x) = 0 in (0, +∞) × ∂Ω,

u(0, x) = u ₀ (x), u _t (0, x) = u ₁ (x) x ∈ Ω,

(4)

where h : Ω → R may have negative values, but in a set which is small if compared

to the set where it is positive. In this way the authors generalize previous results

in the 1–dimensional case by Freitas–Zuazua ([4]) and Benaddi–Rao ([1]). But of

course, here the damping depends on the spatial variable, and not on the time

variable, as it happens in some applications in aerodynamics and mesodynamics

which we have in mind. For example, nose wheel shimmy of an airplane is the

consequence of a negative damping, which is controlled by a suitable hydraulic

shimmy damper which induces a positive damping ([17]). Another example of

sign–changing damping comes from Quantum Field Theory and Landau instability

(see [7]) and from mesodynamics with the laser driven pendulum, which is a long

period pendulum forced by external means into a ”far-from-equilibrium” condition

by anisotropic friction (so that the damping changes sign, see [2]). En passant,

we recall that negative damping may appear in every–day–life, for example Gunn

diodes, used as source of microwave power, and suspension bridges ([8], [14], [11],

[12]), which may experience negative damping in a catastrophic way, like Takoma

Bridge.

To our best knowledge, this paper contains the first stability results for solutions of nonlinear damped wave equations with time–dependent sign–changing damping.

In particular, as already said, we are interested in a nonlinear function f whose prototype is f (u) = −|u| ^p u. We recall that such a model was already considered in [16] for systems of ordinary differential equations. However, it is impossible to compare our results with previous ones contained in the papers cited so far, since up to now only nonnegative dampings have been considered for nonlinear partial differential equations.

2. The abstract setting. Having in mind (1), we will use the following abstract setting: let us consider a second order evolution equation of the form

u ⁰⁰ + B(t)u ⁰ + Au = f (u) (5)

and its associated Cauchy problem

 u ⁰⁰ + B(t)u ⁰ + Au = f (u) t > 0

u(0) = u 0 ∈ V, u ⁰ (0) = u 1 ∈ H. (6) Here H denotes a real Hilbert space with scalar product h·, ·i H and norm k · k H , A : D(A) ⊂ H → H is a linear self–adjoint coercive operator on H with dense domain, and V = D(A ^1/2 ) with norm kvk V = kA ^1/2 vk H is such that

V ,→ H ≡ H ⁰ ,→ V ⁰

with dense embeddings, so that there exists λ 1 > 0 such that

λ ₁ kuk ² _H ≤ kuk ² _V ∀ u ∈ V. (7) Of course, in the model case H = L ² (Ω), V = H ₀ ¹ (Ω) and λ 1 is the first eigenvalue of −∆ on H ₀ ¹ (Ω), so that the previous inequality represents the classical Poincar´ e inequality.

By solution of (6), we mean a function u such that for any T > 0 there holds u ∈ L ² (0, T ; V ) ∩ H ¹ (0, T ; H) ∩ H ² (0, T ; V ⁰ )

with B(t)u ⁰ (t) ∈ H for any t, hBu ⁰ , u ⁰ i H ∈ L ² (0, T ),

Au ∈ L ² (0, T ; V ⁰ ), Bu ⁰ ∈ L ² (0, T ; V ⁰ ), f (u) ∈ L ² (0, T ; H), with u(0) = u 0 , u ⁰ (0) = u 1 , and such that

u ⁰⁰ + Bu ⁰ + Au = f (u) in L ² (0, T ; V ⁰ ).

Remark 1. In the case of problem (1) the condition f (u) ∈ L ² (0, T ; H) is obviously satisfied when H = L ² (Ω), V = H ₀ ¹ (Ω) and f (u) = −|u| ^p u, p ≥ 0 if N = 1, 2 or 0 ≤ p ≤ 4/(N − 2) if N ≥ 3.

We assume that the problem is variational, i.e. there exists a real–valued func- tional F such that F (0) = 0 (just for the sake of simplicity) and F ⁰ (u)(φ) = hf (u), φi V

,V for any u, φ ∈ V . Of course, in problem (1) we have

F (u) = Z

Ω

F (u) dx, where F (s) = R s

0 f (t) dt, i.e. F (s) = − ^|s| _p+2

for the model case.

For any solution u of problem (6) we denote by E u , or simply by E if there is no confusion, the energy of such a solution, i.e.

E(t) = 1

2 ku ⁰ (t)k ² _H + 1

2 ku(t)k ² _V dx − F (u(t)). (8)

Finally, we assume that B ∈ L ^∞ (0, T ; Lip(H, H ⁰ )) for any T > 0, that B(t) is a bounded operator and B(t)0 = 0 for any t ≥ 0, and that f is continuous. Under the same assumption, when f ≡ 0, in [5] the result established in the following lemma is given; it still holds true in the nonlinear case thanks to the assumption f (u) ∈ L ² (0, T ; H). The proof is an adaptation of the one given therein and is thus omitted.

Lemma 2.1. For any solution u of (6) we have

• u ∈ C([0, T ]; V ) ∩ C ¹ ([0, T ]; H);

• the associated energy E _u is locally absolutely continuous on [0, ∞) and E _u ⁰ (t) = −hB(t)u ⁰ (t), u ⁰ (t)i H a.e. in [0, ∞). (9) We will also need the following obvious corollary.

Lemma 2.2. If hB(t)w, wi H ≥ 0 (resp. ≤ 0) for a.e. t in a given interval I and for every w ∈ H, then for any solution u of (6) the associated energy E u is non increasing (resp. decreasing) in I.

Remark 2. In the case of problem (1) equality (9) reads E ⁰ (t) = −

Z

Ω

h(t)u ² _t dx,

which is non positive in an interval I if h ≥ 0 a.e. in I and non negative if h ≤ 0 a.e. in I.

3. The positive–negative damping. As already said, concerning the nonlinear- ity f , our prototype is the function f (u) = −|u| ^p u, p ≥ 0, which obviously satisfies

f (s)s − F (s) ≤ 0 ∀ s ∈ R. (10)

We also remark that this sign assumption on f seems quite reasonable and hard to relax. Indeed, it is well known that solutions of u _tt + a(x, t)u _t − ∆u = |u| ^p u in Ω, a(x, t) ≥ 0 and p > 0, may blow up in finite time (see, for example, [9]), so that asymptotic stability is a senseless requirement.

The inequality expressed in (10), in the abstract case is replaced by the condition that

hf (u), ui V

,V − F (u) ≤ 0 ∀ u ∈ V, (11) though it would suffice to hold only for those u’s which solve (6). Of course this condition, in effect, reduces to (10) in the model case of problem (1).

Finally, we consider a time–dependent operator B with the property that B ∈ L ^∞ _loc (R ⁺ , Lip(H)), and more precisely we focus on a positive–negative B. For this kind of damping we consider, as in [5], a sequence of open and disjoint intervals I 2n := (a 2n , a 2n+1 ) and I 2n+1 := (a 2n+1 , a 2n+2 ) of (0, ∞) with the property that a n < a n+1 for any n ∈ N and a ⁿ → ∞ as n → ∞.

We make the following assumption:

Hypothesis 3.1. For all n ∈ N, there exist three positive constants, m ²ⁿ , M 2n , and M 2n+1 , such that

m _2n kvk ² _H ≤ hB(t)v, vi H ∀ t ∈ I 2n , ∀ v ∈ H, (12) kB(t)vk ² _H ≤ M 2n hB(t)v, vi H ∀ t ∈ I 2n , ∀ v ∈ H, (13)

−M 2n+1 kvk ² _H ≤ hB(t)v, vi H ≤ 0 ∀ t ∈ I 2n+1 , ∀ v ∈ H. (14)

The previous assumption means that B is a positive operator on I 2n and a negative operator on I 2n+1 . Thus, by Lemma 2.1, the energy decays on the time intervals I 2n and increases on the time intervals I 2n+1 . Indeed, in the model case this situation corresponds to a damping h which is strictly positive in I 2n and strictly negative in I 2n+1 .

In order to prove our main result we recall the following theorem.

Theorem 3.1 (see [3], Theorem 4.1 and Remark 4.1). Fix b > a ≥ 0 and assume there exist M ≥ m > 0 such that

mkvk ² _H ≤ hB(t)v, vi H ∀ t ∈ [a, b], ∀ v ∈ H (15) kB(t)vk ² _H ≤ M hB(t)v, vi H ∀ t ∈ [a, b], ∀ v ∈ H. (16) Moreover, suppose that (7) and (11) hold. Then any solution u of equation (5) satisfies

E u (b) ≤ 1

1 + T ³ 30

1

4

λ

m + _32m ^3T

+ ^{M T} _16λ

1

E u (a), (17)

where we have set T := b − a.

We recall that in the model case with f (s) = −|s| ^p s, it results F (s) ≤ 0 for any s, so that also F (s) ≤ 0. More generally, this condition is satisfied if sf (s) ≤ 0 for any s; this condition is again satisfied by the model case, and therefore we require this assumption also in the general setting of our next result, the main theorem of the paper.

For the sake of simplicity, now we set T n = length of I n .

Theorem 3.2. Assume (7), (11) and Hypothesis 3.1. Moreover assume that F ≤ 0 and that

∞

X

p=0



2M 2p+1 T 2p+1 − log



1 + T _2p ³ 30

1

4

λ

m

+ ^3T

2 2p

32m

+ ^M

^T

2 2p

16λ







 = −∞. (18) Then for every (u 0 , u 1 ) ∈ V × H, if u solves problem (6), then u is asymptotically stable, i.e. E u (t) → 0 as t → ∞.

Proof. For all n ∈ N, applying Theorem 3.1 on I _2n , we obtain

E u (a 2n+1 ) ≤ 1

1 + T _2n ³ 30

1

4

λ

m

+ _32m ^3T

2n

+ ^M _16λ

^T

1

E u (a 2n ). (19)

On the other hand on I 2n+1 , by (9), (14), (8) and using the fact that F ≤ 0, we have

0 ≤ E ⁰ _u (t) = −hB(t)u ⁰ (t), u ⁰ (t)i H ≤ M 2n+1 ku ⁰ (t)k ² _H ≤ 2M 2n+1 E u (t).

Thus

E u (a 2n+2 ) ≤ E u (a 2n+1 )e ^2M

^T

. Using (19), one has

E u (a 2n+2 ) ≤







n

Y

p=0

e ^2M

^T



1 + T _2p ³ 30

1

4

λ

m

+ ^3T

2 2p

32m

+ ^M

^T

2 2p

16λ





−1 



 E u (0).

Hence if

∞

Y

p=0

e ^2M

^T



1 + T _2p ³ 30

1

4

λ

m

+ _32m ^3T

2p

+ ^M _16λ

^T

1





−1

= 0,

then we obtain the stability result. But this is equivalent to (18), and thus E _u (t) → 0 as t → ∞.

Example 1. A simple but interesting case is to consider the uniformly distributed damping, that is M _2n = m _2n , M _2n+1 = M and T _2n = T ; an easy calculation shows that condition (18) is satisfied if P T 2n+1 < ∞.

Remark 3. We wish to underline again the fact that (18) is a sufficient condition for stability and that it is a condition on the lengths of the time intervals where the damping is positive or negative and the maximal and minimal values of the damping in those intervals. In particular we remark that the distribution of those intervals has no relevance. Therefore, the result of the present paper, together with the result of [5], shows that, in some sense, the “uniformly damped” wave equation behaves like ordinary differential equations. This is opposite to the case of the wave equation subject to a “locally distributed” damping or to a boundary damping, as shown in [13], where the authors prove that the distribution of the intervals plays a crucial rˆ ole, due to the optic rays propagation. This facts together seem to suggest that the possibility to extend known results from ODE’s to PDE’s strongly relies in considering a uniformly distributed damping.

4. Some applications. A particular case of the abstract problem considered in the Section 3 is the following nonlinear wave system in a bounded domain Ω of R ^N , N ≥ 1:

(W )







u _tt = ∆u − h(t)g(u _t ) + f (u) in (0, +∞) × Ω,

u(t, x) = 0 in (0, +∞) × ∂Ω,

u(0, x) = u ₀ (x), u _t (0, x) = u ₁ (x) x ∈ Ω, where the following assumptions are made:

(A)



 



 



g : R −→ R is a C ¹ function with g(0) = 0,

∃ B ≥ A > 0 such that 0 < A ≤ g ⁰ (v) ≤ B ∀ v ∈ R, f satisfies (10),

sf (s) ≤ 0 for any s ∈ R, while u 0 ∈ H ₀ ¹ (Ω) and u 1 ∈ L ² (Ω).

Indeed, with the assumption above, the term h(t)u t can be easily replaced by the nonlinear term h(t)g(u t ). In this case Theorems 3.1 and 3.2 read respectively as follows.

Theorem 4.1. Fix b > a ≥ 0 and assume that there exist M ≥ m > 0 such that 0 < m ≤ h(t) ≤ M ∀ t ∈ [0, T ]. (20) Suppose that (A) holds. Then every solution of

u tt = ∆u − h(t)g(u t ) + f (u) verifying homogeneous Dirichlet boundary conditions, satisfies

E(b) ≤ 1 + T ³ 30

1

4

λ

+ _32m ^3T

+ ^{M T} _16λ

1

! −1

E(a).

Theorem 4.2. Let (a n , b n ) n be a sequence of disjoint open intervals in (0, +∞) with a n → +∞ and assume that Hypothesis 3.1, conditions (18) and (A) hold true.

Then for every (u 0 , u 1 ) ∈ H ₀ ¹ (Ω) × L ² (Ω) the solution u of problem (W ) is such that E u (t) → 0 as t → +∞.

Of course the abstract setting we gave in Section 2 lets us deal with higher order problem in a bounded and smooth domain Ω ⊂ R ^N . For instance, consider the following evolution problem in presence of a polyharmonic operator:

(H)







u tt = ∆ ^2m u − h(t)g(u t ) + f (u) in (0, +∞) × Ω,

Cu(t, x) = 0 ∈ R ^2m in (0, +∞) × ∂Ω,

u(0, x) = u 0 (x) ∈ D(∆ ^m ), u t (0, x) = u 1 (x) x ∈ Ω,

where m ∈ N, g and f are as before and C is a boundary operator such that (B)

 the first eigenvalue of ∆ ^2m under the boundary conditions Cu(t, x) = 0 ∈ R ^2m in (0, +∞) × ∂Ω is strictly positive.

For example, if m = 1 and C denotes the Dirichlet operator, that is Cu = (u, ∂u/∂ν), ν being the unit outward normal to ∂Ω, we have D(∆) = H ₀ ² (Ω), while, in the case of Navier conditions, i.e. Cu = (u, ∆u), we have D(∆) = H ₀ ¹ (Ω)∩H ² (Ω).

Other generalization, in the case m > 1, are now easy and straightforward to do, obviously excluding the case of Neumann conditions, for which the first eigenvalue is 0. For the other cases we can prove the following result, which is a particular version of Theorem 3.2.

Theorem 4.3. Let (a _n , b _n ) _n be a sequence of disjoint open intervals in (0, +∞) with a _n → +∞ and assume that Hypothesis 3.1 and conditions (18) and (A) − (B) hold true. Then for every (u 0 , u 1 ) ∈ D(∆ ^m ) × L ² (Ω) the solution u of problem (H) is such that E u (t) → 0 as t → +∞.

REFERENCES

[1] A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping, J.

Differential Equations 161 (2000), no. 2, 337357.

[2] C.W. de Silva, “Vibration and Shock Handbook”, Mechanical Engineering, CRC Press 2005.

[3] G. Fragnelli and D. Mugnai, Stability of solutions for some classes ofnonlinear damped wave equations, SIAM J. Control Optim. 47 (2008), no. 5, 2520–2539.

[4] P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping, J.

Differential Equations 132 (1996), no. 2, 338352.

[5] A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently con- trolled second order evolution equations, SIAM J. Control and Opt., 43 (2005), no. 6, 2089–

2108.

[6] L. Hatvani and T. Krisztin, Necessary and sufficient conditions for intermittent stabilization of linear oscillators by large damping, Differential Integral Equations 10 (1997), no. 2, 265–

272.

[7] S. Konabe and T. Nikuni, Coarse–Grained Finite–Temperature Theory for the Bose Con- densate in Optical Lattices, J. Low Temp. Phys. (2008) 150, 12-46.

[8] A.C. Lazer and P.J. McKenna, Large–amplitude periodic oscillations in suspension bridges:

some new connections with nonlinear analysis, SIAM Review 32 (1990), 537–578.

[9] H.A. Levine, S.R. Park and J. Serrin, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl. 228 (1998), no. 1, 181–205.

[10] K. Liu, B. Rao and X. Zhang, Stabilization of the wave equations with potential and indefinite damping, J. Math. Anal. Appl. 269 (2002), no. 2, 747–769.

[11] A. Marino and D. Mugnai, Asymptotically critical points and their multiplicity, Topol. Meth-

ods Nonlinear Anal. (2002) Vol. 19, No. 1, pp. 29–38.

[12] A. Marino and D. Mugnai, Asymptotical multiplicity and some reversed variational inequal- ities, Topol. Methods Nonlinear Anal. (2002) Vol. 20. No. 1, pp. 43–62.

[13] P. Martinez and J. Vancostenoble, Stabilization of the wave equation by on–off and positive–

negative feedbacks, ESAIM Control Optim. Calc. Var. 7 (2002), 335–377.

[14] D. Mugnai, On a “reversed” variational inequality, Topol. Methods Nonlinear Anal. (2001) 17, No. 2, pp. 321–358.

[15] P. Pucci and J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscilla- tors, SIAM J. Math. Anal. 25 (1994), no. 3, 815–835.

[16] P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for non- linear second order systems. II, J. Differential Equations 113 (1994), no. 2, 505–534.

[17] G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods, Aerosp. Sci. Technol. 1 (1997), no. 8, 545–555.

E-mail address: fragnelli@dii.unisi.it

E-mail address: mugnai@dmi.unipg.it